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Chapter 2: Problem 47

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$

Short Answer

Expert verified
Answer: The vertical asymptote of the function is at $$x=-1$$.

Step by step solution

01

Understand the function

The given function is $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$. To find the vertical asymptotes, we'll focus on the denominator, \((x+1)^3\).
02

Find the values of x for which the denominator is equal to 0

To find the values of x where the denominator is equal to 0, we'll set the denominator to 0 and solve for x: $$(x+1)^3 = 0$$ To find the cube root of both sides, $$x+1=0$$ Then, we will subtract 1 from both sides: $$x=-1$$
03

Identify the vertical asymptotes

The function $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$ has a vertical asymptote at $$x=-1$$, since this is where the denominator equals 0. To verify this, you can use a graphing utility to plot the function and see that the graph approaches infinity as x approaches -1.

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