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Chapter 4: Problem 64

Evaluate the following limits or explain why they do not exist. Check your results by graphing. $$\lim _{x \rightarrow 0}\left(2^{a x}+x\right)^{1 / x}, \text { for a constant } a$$

Short Answer

Expert verified
Answer: The limit of the function as \(x\) approaches 0 is \((2^a)e\).

Step by step solution

01

Factor out the constant

Since \(a\) is a constant, we can factor it out from the function in the exponent: $$\lim _{x \rightarrow 0}(2^{ax}+x)^{1/x} = \lim _{x \rightarrow 0}(2^{ax})^{1/x}(1+x)^{1/x}$$
02

Use the property of exponents

Now, use the property of exponents that states: \((a^b)^c = a^{bc}\), so we can simplify the expression: $$\lim _{x \rightarrow 0}(2^{ax})^{1/x}(1+x)^{1/x} = \lim _{x \rightarrow 0}(2^{ax/x})(1+x)^{1/x}$$
03

Simplify the expression further

We can simplify the expression even more as follows: $$\lim _{x \rightarrow 0}(2^a)(1+x)^{1/x}$$ Since the limit of product is equal to the product of limits: $$\lim _{x \rightarrow 0}(2^a)(1+x)^{1/x} = (2^a) \lim _{x \rightarrow 0}(1+x)^{1/x}$$
04

Evaluate the limit of \((1+x)^{1/x}\)

Now we focus on evaluating the limit: \(\lim _{x \rightarrow 0}(1+x)^{1/x}\). This limit has a well-known result: $$\lim _{x \rightarrow 0}(1+x)^{1/x} = e$$
05

Multiply the constants

Multiply the constants, to find the final answer: $$(2^a)e$$
06

Graphing the function (Optional)

Graph the function \((2^{ax}+x)^{1/x}\) for different values of the constant \(a\). In every case, you will observe that as \(x\) approaches 0, the function value approaches \((2^a)e\).

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