Solved: "Evaluate the following limits. limx→∞(2+4x)2x"

Burhan Hopper

Answered question

2021-10-29

Evaluate the following limits.

lim_{x \to \infty} {(2 + \frac{4}{x})}^{2 x}

comentezq

Skilled2021-10-30Added 106 answers

Given:

lim_{x \to \infty} {(2 + \frac{4}{x})}^{2 x}

Calculation:

lim_{x \to \infty} {(2 + \frac{4}{x})}^{2 x}

if we substitute limit

x \to \infty

in this we get indeterminate form

2^{\infty}

Let's define

y = {(2 + \frac{4}{x})}^{2 x}

And take the log on both side and we will do some simplification

\ln y = {\ln (2 + \frac{4}{x})}^{2 x}

And take the log on both side and we will do some simplification

\ln y = {\ln (2 + \frac{4}{x})}^{2 x}

\Rightarrow \ln y = 2 x \ln (2 + \frac{4}{x})

\ln y = \frac{\ln (2 + \frac{4}{x})}{\frac{1}{2 x}}

Taking limit

x \to \infty

on both side

lim_{x \to \infty} [\ln y] = \frac{\ln (2 + \frac{4}{x})}{\frac{1}{2 x}}

lim_{x \to \infty} [\ln y] = \frac{\ln (2 + \frac{4}{\infty})}{\frac{1}{2 \infty}}

lim_{x \to \infty} [\ln y] = \frac{\ln (2 + 0)}{0} = \frac{\ln (2)}{0} = \infty

We know that

t = e^{\ln y}

This means we can do original limit as follow

lim_{x \to \infty} {(2 + \frac{4}{x})}^{2 x} = lim_{x \to \infty} [y] = lim_{x \to \infty} e^{\ln y} = e^{lim_{x \to \infty} [\ln y]} e^{\infty} = \infty

Therefore

lim_{x \to \infty} {(2 + \frac{4}{x})}^{2 x} = \infty

Jeffrey Jordon

Expert2022-08-30Added 2605 answers

Answer is given below (on video)

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