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Problem 6. Compute the following: a) $\lim_{x \to 0} x \left( \int_0^x \cos\left(\frac{1}{t+1}\right) dt \right)$, b) $\lim_{x \to 0^+} \frac{\tan x - x}{x^3}$, $x \in \left(0, \frac{\pi}{2}\right)$, c) $\lim_{x \to 0^+} xe^{\left(\frac{1}{x}\right)}$.

          Problem 6. Compute the following:
a) $\lim_{x \to 0} x \left( \int_0^x \cos\left(\frac{1}{t+1}\right) dt \right)$, b) $\lim_{x \to 0^+} \frac{\tan x - x}{x^3}$, $x \in \left(0, \frac{\pi}{2}\right)$, c) $\lim_{x \to 0^+} xe^{\left(\frac{1}{x}\right)}$.

Problem 6. Compute the following:
a) limx → 0 x ( ∫0^x cos((1)/(t+1)) dt ), b) limx → 0^+(tan x - x)/(x^3), x ∈(0, (π)/(2)), c) limx → 0^+ xe^((1)/(x)).

Added by Iker L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/18/2023

Step 1

Since the integrand cos(\frac{1}{t+1}) does not have an elementary antiderivative, we cannot directly compute the integral. However, we can use the Fundamental Theorem of Calculus to find the derivative of the integral with respect to x. Let F(x) = \int_0^x Show more…

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Compute the following (with proof as this is for real analysis): a) lim_(x->0)x(int_0^x cos((1)/(t+1))dt), b) lim_(x->0^(+))(tanx-x)/(x^(3)),xin(0,(pi )/(2)), c) lim_(x->0+)xe^(((1)/(x))). Problem 6. Compute the following: tan x-x a) lim x cos( dt b) lim ,xe(0,),c) lim xe() -0 0+ x3