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Numerade Educator

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Problem 69 Hard Difficulty

Use a graph to estimate the value of the limit. Then use l'Hospital's Rule to find the exact value.

$ \displaystyle \lim_{x\to \infty} \left( 1 + \frac{2}{x} \right)^x $

Answer

$$\begin{array}{l}
\text { Now } y=\left(1+\frac{2}{x}\right)^{x} \Rightarrow \ln y=x \ln \left(1+\frac{2}{x}\right) \Rightarrow \\
\begin{aligned}
\lim _{x \rightarrow \infty} \ln y &=\lim _{x \rightarrow \infty} \frac{\ln (1+2 / x)}{1 / x}=\lim _{x \rightarrow \infty} \frac{\frac{1}{1+2 / x}\left(-\frac{2}{x^{2}}\right)}{-1 / x^{2}} \\
=& 2 \lim _{x \rightarrow \infty} \frac{1}{1+2 / x}=2(1)=2 \Rightarrow
\end{aligned}
\end{array}$$

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Video Transcript

for this problem, we are going to be using a graph to estimate the value of the limits. So in this case our limit, our function is going to be one plus two over X to the power of X. This is our graph and we're looking as the graph approaches infinity. So we want to go over here, we can zoom out a little bit and zoom in further over here to see what value our graph is approaching um and it's going down but it appears to be approaching 7.3, repeating, but we can't know for sure until we actually evaluate the limit. Um We can use to help the taliban rule though and when we use the hotel's role after taking the natural log of both sides where we end up getting um to simplify it to is the natural log of one plus two over X, divided by one over X. This allows us to write T as one over X. So we can change this to to team. Then we can evaluate this, we get the indeterminant form. But once we differentiate the top and the bottom separately and evaluated at X, going to infinity, we end up getting um two as our answer. So that is actually E to the two. And as we see, that's very close to the value is approaching beforehand. So we see that, wow the limit may not tell us exactly. It's good at approximating if we just look at the graph.