A) Find the limit: limit as x approaches infinity of arctan(e^x). B) Evaluate the integral:...


A) Find the limit: {eq}\; \lim_{x\rightarrow \, \infty} \arctan(e^x) {/eq}.

B) Evaluate the integral: {eq}\int_{0}^{ \frac{\sqrt{3}}{5} } \, \frac{\mathrm{d}x}{1 \, + \, 25x^2} {/eq}.

The Limit and The Integral in Calculus:

To solve the first problem, we'll use the limit chain rule. The limit chain rule states that: If {eq}\displaystyle \lim_{u\rightarrow b} f(u) = L , \lim_{x\rightarrow a} g(x) = b {/eq} and {eq}f(x) {/eq} is continuous at {eq}x= b {/eq} then {eq}\displaystyle \lim_{u\rightarrow a} f\left(g(x) \right)= L \\ {/eq}

To solve the second problem, we'll use the integral substitution and compute the boundaries.

Answer and Explanation:


We are given:

{eq}\displaystyle \lim_{x\rightarrow \, \infty} \arctan(e^x) {/eq}

Apply limit chain rule:

where {eq}g(x) =e^x , f(u)=...

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from PSAT Prep: Tutoring Solution

Chapter 10 / Lesson 13

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