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Chapter 4: Problem 66

Evaluate the following limits or explain why they do not exist. Check your results by graphing. $$\lim _{z \rightarrow \infty}\left(1+\frac{10}{z^{2}}\right)^{z^{2}}$$

Short Answer

Expert verified
Answer: The limit of the given expression as z approaches infinity is $$e^{10}$$.

Step by step solution

01

Rewrite the Expression

We can rewrite the given limit expression by substituting \(w=10/z^{2}\). So we get, $$\lim _{z \rightarrow \infty}\left(1+\frac{1}{w}\right)^{10} = \left(\lim _{z \rightarrow \infty}\left(1+\frac{1}{w}\right)\right)^{10}$$
02

Solving the Limit

Now the expression becomes the limit of the base raised to the 10th power. Use the fact that $$\lim_{w\rightarrow 0}(1+w)^{\frac{1}{w}} = e$$ In this case, as \(z\rightarrow \infty\), \(w\) approaches \(0\). Then, we get the limit of the base to be \(e\). Therefore, the expression becomes $$e^{10}$$ So the limit of the given function is: $$\lim _{z \rightarrow \infty}\left(1+\frac{10}{z^{2}}\right)^{z^{2}} = e^{10}$$

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