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13

Q1.

Solution

Q2.

Solution

limit as x rightwards arrow 1 of open parentheses log subscript 5 space 5 x close parentheses to the power of log subscript x 5 end exponent
equals limit as x rightwards arrow 1 of open parentheses log subscript 5 5 plus log subscript 5 x close parentheses to the power of log subscript x 5 end exponent
equals limit as x rightwards arrow 1 of open parentheses 1 plus log subscript 5 x close parentheses to the power of log subscript x 5 end exponent
equals limit as x rightwards arrow 1 of open parentheses 1 plus log subscript 5 x close parentheses to the power of fraction numerator 1 over denominator log subscript 5 x end fraction end exponent
I f space x rightwards arrow 1 comma space t h e n space log subscript 5 x rightwards arrow 0
L e t space log subscript 5 x equals y
equals limit as y rightwards arrow 0 of open parentheses 1 plus y close parentheses to the power of 1 over y end exponent
equals e to the power of limit as y rightwards arrow 0 of 1 over y log left parenthesis 1 plus y right parenthesis end exponent
to the power of limit as y rightwards arrow 0 of 1 over y log left parenthesis 1 plus y right parenthesis end exponent equals 1 space s o
e to the power of limit as y rightwards arrow 0 of 1 over y log left parenthesis 1 plus y right parenthesis end exponent equals e
Q3.

Solution

Q4.

Solution

Q5. Find the derivative of 3 tan x + 5 log x + 5ex

Solution

Q6.

Solution

Q7.

Solution

When we put x = 2, then the expression does not take an indeterminate form and hence the answer is obtained by substituting x = 2
Q8.

Solution

Q9.

Solution

Q10.

Solution

Q11.

Solution

Q12. P r o v e space t h a t space limit as n rightwards arrow infinity of open parentheses 4 to the power of n plus 5 to the power of n close parentheses to the power of 1 over n end exponent equals 5

Solution

limit as n rightwards arrow infinity of open parentheses 4 to the power of n plus 5 to the power of n close parentheses to the power of 1 over n end exponent
equals limit as n rightwards arrow infinity of open parentheses 5 to the power of n close parentheses to the power of 1 over n end exponent open parentheses 1 plus open parentheses 4 over 5 close parentheses to the power of n close parentheses to the power of 1 over n end exponent
equals limit as n rightwards arrow infinity of open parentheses 5 to the power of n close parentheses to the power of 1 over n end exponent open parentheses 1 plus open parentheses 4 over 5 close parentheses to the power of n close parentheses to the power of open parentheses 5 over 4 close parentheses to the power of n.1 over n. open parentheses 4 over 5 close parentheses to the power of n end exponent
L e t space open parentheses 4 over 5 close parentheses to the power of n equals z. space I f space n rightwards arrow infinity comma space t h e n space z rightwards arrow 0.
equals 5. limit as z rightwards arrow 0 of open parentheses open parentheses 1 plus z close parentheses to the power of z close parentheses to the power of 0.0 end exponent
equals 5. e to the power of 0.0 end exponent equals 5
Q13.

Solution

Q14.

Solution

limit as x rightwards arrow a of fraction numerator x to the power of begin display style 3 over 5 end style end exponent minus a to the power of 3 over 5 end exponent over denominator x to the power of begin display style 1 third end style end exponent minus a to the power of begin display style 1 third end style end exponent end fraction
equals limit as x rightwards arrow a of fraction numerator begin display style fraction numerator x to the power of 3 over 5 end exponent minus a to the power of 3 over 5 end exponent over denominator x minus a end fraction end style over denominator begin display style fraction numerator x to the power of 1 third end exponent minus a to the power of 1 third end exponent over denominator x minus a end fraction end style end fraction
U sin g space t h e space r e s u l t colon limit as x rightwards arrow a of fraction numerator x to the power of n minus a to the power of n over denominator x minus a end fraction equals n a to the power of n minus 1 end exponent space w e space c a n space w r i t e
limit as x rightwards arrow a of fraction numerator begin display style fraction numerator x to the power of 3 over 5 end exponent minus a to the power of 3 over 5 end exponent over denominator x minus a end fraction end style over denominator begin display style fraction numerator x to the power of 1 third end exponent minus a to the power of 1 third end exponent over denominator x minus a end fraction end style end fraction equals fraction numerator begin display style limit as x rightwards arrow a of fraction numerator x to the power of 3 over 5 end exponent minus a to the power of 3 over 5 end exponent over denominator x minus a end fraction end style over denominator begin display style limit as x rightwards arrow a of fraction numerator x to the power of 1 third end exponent minus a to the power of 1 third end exponent over denominator x minus a end fraction end style end fraction equals fraction numerator begin display style 3 over 5 end style a to the power of negative begin display style 2 over 5 end style end exponent over denominator begin display style 1 third end style a to the power of negative begin display style 2 over 3 end style end exponent end fraction equals 9 over 5 a to the power of 4 over 15 end exponent
Q15. Find derivative of the function f(x) = 5x2 + 4x - 2 at x = -4.

Solution

f apostrophe left parenthesis x right parenthesis equals 5 fraction numerator d over denominator d x end fraction left parenthesis x squared right parenthesis plus 4 fraction numerator d over denominator d x end fraction left parenthesis x right parenthesis minus 0
space space space space space space space space equals 5 left parenthesis 2 x right parenthesis plus 4.1 minus 0
space space space space space space space space equals 10 x plus 4
f apostrophe left parenthesis minus 4 right parenthesis equals 10 left parenthesis minus 4 right parenthesis plus 4
equals minus 36
Q16.

Solution

limit as x rightwards arrow 0 of fraction numerator sin squared space a x over denominator sin squared space b x end fraction
equals limit as x rightwards arrow 0 of fraction numerator sin squared space a x. left parenthesis b x right parenthesis squared over denominator left parenthesis a x right parenthesis squared. sin squared space b x end fraction. open parentheses fraction numerator a x over denominator b x end fraction close parentheses squared
equals limit as x rightwards arrow 0 of open parentheses fraction numerator sin space a x over denominator a x end fraction close parentheses squared. limit as x rightwards arrow 0 of open parentheses fraction numerator b x over denominator sin space b x end fraction close parentheses squared. open parentheses fraction numerator a x over denominator b x end fraction close parentheses squared
equals 1.1. open parentheses a over b close parentheses squared open square brackets because limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction equals 1 close square brackets
equals a squared over b squared
Q17.

Solution

Q18.

Solution

Q19. F i n d space t h e space v a l u e space o f space limit as x rightwards arrow infinity of open parentheses sin 1 over x plus cos 1 over x close parentheses to the power of x

Solution

limit as x rightwards arrow infinity of open parentheses sin space 1 over x plus cos space 1 over x close parentheses to the power of x
L e t space y equals 1 over x. space S o space x rightwards arrow infinity comma space t h e n space y rightwards arrow 0
equals limit as y rightwards arrow 0 of open parentheses sin space y plus cos space y close parentheses to the power of 1 over y end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator log space open parentheses sin space y plus cos space y close parentheses over denominator y end fraction space space space open square brackets 0 over 0 space f o r m close square brackets end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator begin display style fraction numerator left parenthesis cos space y minus sin space y right parenthesis over denominator open parentheses sin space y plus cos space y close parentheses end fraction end style over denominator 1 end fraction open square brackets L apostrophe H o s p i t a l apostrophe s space R u l e space close square brackets space space end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator left parenthesis cos space y minus sin space y right parenthesis over denominator open parentheses sin space y plus cos space y close parentheses end fraction end exponent
equals e to the power of limit as y rightwards arrow 0 of fraction numerator left parenthesis 1 minus 0 right parenthesis over denominator open parentheses 0 plus 1 close parentheses end fraction end exponent equals e to the power of 1 equals e
Q20.

Solution

limit as x rightwards arrow 0 of open parentheses fraction numerator tan space x over denominator x end fraction close parentheses to the power of 1 over x squared end exponent
equals e to the power of limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction end exponent space space open square brackets stack lim space with x rightwards arrow 0 below f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent space equals e to the power of limit as x rightwards arrow 0 of g left parenthesis x right parenthesis. log open square brackets f left parenthesis x right parenthesis close square brackets end exponent close square brackets
equals e to the power of limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction end exponent
L e t space A equals to the power of limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction space space space open square brackets 0 over 0 space f o r m close square brackets end exponent
limit as x rightwards arrow 0 of fraction numerator log open parentheses begin display style fraction numerator tan space x over denominator x end fraction end style close parentheses over denominator x squared end fraction space left square bracket L apostrophe space H o s p i t a l apostrophe s space R u l e right square bracket
equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator up diagonal strike x over denominator tan space x end fraction end style. begin display style fraction numerator x space s e c squared space x minus tan space x over denominator x to the power of up diagonal strike 2 end exponent end fraction end style over denominator 2 x end fraction
equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator tan space x end fraction end style. begin display style fraction numerator x space s e c squared space x minus tan space x over denominator x end fraction end style over denominator 2 x end fraction space
equals limit as x rightwards arrow 0 of fraction numerator begin display style x space s e c squared end style x begin display style minus tan space x end style over denominator 2 x squared tan space x end fraction space open square brackets 0 over 0 space f o r m close square brackets
equals limit as x rightwards arrow 0 of fraction numerator begin display style space s e c squared end style x begin display style plus x.2 s e c squared x. tan space x end style minus s e c squared x over denominator 4 x. tan space x plus 2 x squared. s e c squared space x end fraction left square bracket L apostrophe space H o s p i t a l apostrophe s space r u l e right square bracket
equals limit as x rightwards arrow 0 of fraction numerator begin display style 2 x. s e c squared x. tan space x end style over denominator 2 x left parenthesis tan space x plus x. s e c squared space x right parenthesis end fraction
equals limit as x rightwards arrow 0 of fraction numerator begin display style s e c squared x. tan space x end style over denominator left parenthesis tan space x plus x. s e c squared space x right parenthesis end fraction space space open square brackets 0 over 0 space f o r m close square brackets
equals limit as x rightwards arrow 0 of fraction numerator begin display style s e c to the power of 4 x plus 2 s e c squared x. tan squared x end style over denominator s e c squared x plus s e c squared space x plus 2 x. s e c squared x. tan space x end fraction open square brackets L apostrophe H o s p i t a l apostrophe s space r u l e close square brackets
equals limit as x rightwards arrow 0 of fraction numerator begin display style s e c squared x left parenthesis 1 plus 2. tan squared x right parenthesis end style over denominator s e c squared space x left parenthesis 2 plus 2 x. tan space x right parenthesis end fraction
equals limit as x rightwards arrow 0 of fraction numerator begin display style left parenthesis 1 plus 2. tan squared x right parenthesis end style over denominator left parenthesis 2 plus 2 x. tan space x right parenthesis end fraction
equals 1 half
therefore A equals 1 half
therefore limit as x rightwards arrow 0 of open parentheses fraction numerator tan space x over denominator x end fraction close parentheses to the power of 1 over x squared end exponent equals e to the power of 1 half end exponent
Q21.

Solution

Q22. Find space limit as straight x rightwards arrow 1 of left parenthesis 19 straight x squared minus 14 straight x right parenthesis space using space algebra space of space limits.

Solution

W e space k n o w space t h a t
stack lim space with straight x rightwards arrow straight a below space left parenthesis straight f left parenthesis straight x right parenthesis minus straight g left parenthesis straight x right parenthesis right parenthesis equals limit as straight x rightwards arrow straight a of straight f left parenthesis straight x right parenthesis minus limit as straight x rightwards arrow straight a of straight g left parenthesis straight x right parenthesis
stack lim space with straight x rightwards arrow 1 below space left parenthesis 19 x squared minus 14 x right parenthesis equals stack lim space with straight x rightwards arrow 1 below 19 x squared minus stack lim space with straight x rightwards arrow 1 below 14 x space
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 19 left parenthesis 1 right parenthesis squared minus 14 left parenthesis 1 right parenthesis equals 5
Q23. Find the derivative of (x2 + 1) cos x

Solution

Q24. Find the derivative of tan x at x = 0

Solution

Q25.

Solution

limit as x rightwards arrow 0 of fraction numerator left parenthesis 1 minus cos 2 x right parenthesis sin 5 x over denominator x squared sin 3 x end fraction
equals limit as x rightwards arrow 0 of fraction numerator 2 sin squared x sin 5 x over denominator x squared sin 3 x end fraction
equals 2 limit as x rightwards arrow 0 of fraction numerator sin squared x over denominator x squared end fraction limit as x rightwards arrow 0 of fraction numerator 5 x open parentheses begin display style fraction numerator sin 5 x over denominator 5 x end fraction end style close parentheses over denominator 3 x open parentheses begin display style fraction numerator sin 3 x over denominator 3 x end fraction end style close parentheses end fraction
equals 2.1.5 over 3.1 equals 10 over 3
Q26.

Solution

limit as x rightwards arrow pi over 2 of open parentheses fraction numerator 1 plus c o t space x over denominator 1 plus cos space x end fraction close parentheses to the power of fraction numerator 1 over denominator cos space x end fraction end exponent
equals limit as x rightwards arrow pi over 2 of open parentheses 1 plus c o t space x close parentheses to the power of fraction numerator 1 over denominator cos space x space end fraction end exponent obelus divided by limit as x rightwards arrow pi over 2 of open parentheses 1 plus cos space x close parentheses to the power of fraction numerator 1 over denominator cos space x end fraction end exponent
L e t space cos space x equals z. space I f space x rightwards arrow pi over 2 comma space t h e n space z rightwards arrow 0
equals limit as x rightwards arrow pi over 2 of open square brackets open parentheses 1 plus c o t space x close parentheses to the power of fraction numerator 1 over denominator begin display style fraction numerator cos space x over denominator sin space x end fraction end style space end fraction. end exponent close square brackets to the power of limit as x rightwards arrow pi over 2 of fraction numerator 1 over denominator sin space x end fraction end exponent obelus divided by limit as z rightwards arrow 0 of open parentheses 1 plus z close parentheses to the power of 1 over z end exponent
L e t space p equals c o t space x. space I f space x rightwards arrow pi over 2 comma space t h e n space p rightwards arrow 0
equals limit as p rightwards arrow 0 of open square brackets open parentheses 1 plus p close parentheses to the power of fraction numerator 1 over denominator begin display style p end style space end fraction end exponent close square brackets obelus divided by limit as z rightwards arrow 0 of open parentheses 1 plus z close parentheses to the power of 1 over z end exponent
equals e obelus divided by e equals 1
Q27. Find the derivative of x2 sin x

Solution

Q28.

Solution

H e r e
fraction numerator d over denominator d x end fraction open parentheses x squared plus 1 over x squared close parentheses cubed
equals fraction numerator d over denominator d x end fraction open parentheses x to the power of 6 plus 1 over x to the power of 6 plus 3 x squared plus 3 over x squared close parentheses
equals fraction numerator d over denominator d x end fraction left parenthesis x to the power of 6 right parenthesis plus fraction numerator d over denominator d x end fraction open parentheses 1 over x to the power of 6 close parentheses plus fraction numerator d over denominator d x end fraction left parenthesis 3 x squared right parenthesis plus fraction numerator d over denominator d x end fraction open parentheses 3 over x squared close parentheses
equals 6 x to the power of 5 minus 6 x to the power of minus 7 end exponent plus 6 x minus 6 x to the power of minus 3 end exponent
Q29.

Solution

Q30.

Solution

Q31.

Solution

limit as x rightwards arrow 0 of fraction numerator 3 to the power of x minus 1 over denominator square root of x plus 1 end root minus 1 end fraction
equals limit as x rightwards arrow 0 of fraction numerator 3 to the power of x minus 1 over denominator x end fraction. fraction numerator x over denominator square root of x plus 1 end root minus 1 end fraction
equals limit as x rightwards arrow 0 of fraction numerator 3 to the power of x minus 1 over denominator x end fraction. limit as x rightwards arrow 0 of fraction numerator x open parentheses square root of x plus 1 end root plus 1 close parentheses over denominator open parentheses square root of x plus 1 end root minus 1 close parentheses open parentheses square root of x plus 1 end root plus 1 close parentheses end fraction
equals log space 3. space limit as x rightwards arrow 0 of fraction numerator x open parentheses square root of x plus 1 end root plus 1 close parentheses over denominator x plus 1 minus 1 end fraction
equals log space 3. space limit as x rightwards arrow 0 of fraction numerator x open parentheses square root of x plus 1 end root plus 1 close parentheses over denominator x end fraction
equals 2 log space 3 equals log space 9
Q32.

Solution

Q33.

Solution

Q34. Find the derivative of x3 + Sin x wrt x.

Solution

Q35.

Solution

Q36.

Solution

Q37.

Solution

Q38.

Solution



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