I need to solve them all andif f(x) = sin" (x²)Lim 0find 2m + nf(x)ƒ'(x)(1 cos x)m= 2√3 2xCOS(-+ cas-¹ (15) + tan6*31+e²x2x¹ (22)dx

Answered: and if f(x) = sin" (x²) Lim 0 find 2m +…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

I need to solve them all

and
if f(x) = sin" (x²)
Lim 0
find 2m + n
f(x)ƒ'(x)
(1 cos x)m
= 2√3

2x
COS
(-+ cas-¹ (15) + tan
6*
3
1+e²x
2x
¹ (22)
dx

Expert Solution

Step 1: simplifying limit

Given that $f open parentheses x close parentheses space equals space sin to the power of n open parentheses x squared close parentheses$

$rightwards double arrow$ $f apostrophe open parentheses x close parentheses space equals space n space sin to the power of n minus 1 end exponent open parentheses x squared close parentheses space cos open parentheses x squared close parentheses space open parentheses 2 x close parentheses$

Formula: $limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction space equals space 1$

$limit as x rightwards arrow 0 of space fraction numerator f open parentheses x close parentheses f apostrophe open parentheses x close parentheses over denominator open parentheses 1 minus cos open parentheses x close parentheses close parentheses to the power of m end fraction$ $equals space limit as x rightwards arrow 0 of space fraction numerator space open square brackets fraction numerator sin to the power of n open parentheses x squared close parentheses over denominator x to the power of 2 n end exponent end fraction x to the power of 2 n end exponent close square brackets. space n space open square brackets fraction numerator sin to the power of n minus 1 end exponent open parentheses x squared close parentheses over denominator x to the power of 2 n minus 2 end exponent end fraction x to the power of 2 n minus 2 end exponent close square brackets space cos open parentheses x squared close parentheses space open parentheses 2 x close parentheses over denominator 2 to the power of m open parentheses fraction numerator sin open parentheses x over 2 close parentheses over denominator x over 2 end fraction open parentheses x over 2 close parentheses close parentheses to the power of 2 m end exponent end fraction$

$equals space limit as x rightwards arrow 0 of space fraction numerator x to the power of 2 n plus 2 n minus 2 plus 1 end exponent space open square brackets fraction numerator sin open parentheses x squared close parentheses over denominator x squared end fraction close square brackets to the power of n. space n space open square brackets fraction numerator sin open parentheses x squared close parentheses over denominator x squared end fraction close square brackets to the power of n minus 1 end exponent space cos open parentheses x squared close parentheses space open parentheses 2 close parentheses over denominator 2 to the power of m open parentheses x over 2 close parentheses to the power of 2 m end exponent open parentheses fraction numerator sin open parentheses x over 2 close parentheses over denominator x over 2 end fraction close parentheses to the power of 2 m end exponent end fraction$

$equals space open square brackets limit as x rightwards arrow 0 of x to the power of 4 n minus 1 end exponent over open parentheses x over 2 close parentheses to the power of 2 m end exponent space close square brackets fraction numerator space open square brackets 1 close square brackets to the power of n. space n space open square brackets 1 close square brackets to the power of n minus 1 end exponent space cos open parentheses 0 close parentheses space open parentheses 2 close parentheses over denominator 2 to the power of m open parentheses 1 close parentheses to the power of 2 m end exponent end fraction$

$equals space open square brackets limit as x rightwards arrow 0 of x to the power of 4 n minus 1 minus 2 m end exponent 2 to the power of 2 m end exponent space close square brackets fraction numerator space 2 n over denominator 2 to the power of m end fraction$

$equals space open square brackets limit as x rightwards arrow 0 of x to the power of 4 n minus 1 minus 2 m end exponent space close square brackets 2 to the power of m plus 1 end exponent n$