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Chapter 4: Problem 56

Evaluate the following limits or explain why they do not exist. Check your results by graphing. $$\lim _{x \rightarrow 0}(1+4 x)^{3 / x}$$

Short Answer

Expert verified
Question: Evaluate the limit as \(x\) approaches \(0\) of \((1+4x)^{\frac{3}{x}}\). Answer: The limit as \(x\) approaches \(0\) of \((1+4x)^{\frac{3}{x}}\) is \(e^{-12}\).

Step by step solution

01

Apply the limit result

First, we rewrite the given limit expression as follows: $$\lim_{x\to 0}\left(\left[\frac{1}{1+4x}\right]^x\right)^{-\frac{3}{x}}$$
02

Evaluate the inner limit

We now evaluate the limit of the inner expression, using the result stated in the analysis: $$\lim_{x\to 0}\left(\frac{1}{1+4x}\right)^x = \lim_{x\to 0}(1+(-4x))^{\frac{1}{-4x}} = e^{-4}$$
03

Substitute the inner limit

Substitute the result of Step 2 back into the original limit expression: $$\lim_{x\to 0}\left[\left(\frac{1}{1+4x}\right)^x\right]^{-\frac{3}{x}} = \left(e^{-4}\right)^{-\frac{3}{x}}$$.
04

Evaluate the final limit

Finally, let's once again apply the important result to the remaining limit expression: $$\lim_{x\to 0}\left(e^{-4}\right)^{-\frac{3}{x}} = e^{3\cdot(-4)} = e^{-12}$$ Thus, the limit as \(x\) approaches \(0\) of \((1+4x)^{\frac{3}{x}}\) is \(e^{-12}\).

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