Question

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

Short answer

Question: Identify the vertical asymptote(s) of the function $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$ using analytical methods and/or a graphing utility. Answer: The function $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$ has one vertical asymptote located at x = -1.

Step-by-step solution

Finding the values of x where the denominator is equal to zero

The first step is setting the denominator to zero, then solving for x: $$(x+1)^{3} = 0$$ Once we have found the x-value(s) responsible for making the denominator zero, we'll be able to determine the location of the vertical asymptotes. 2.

Solve for x

Now we'll solve the equation for x: $$(x+1)^{3} = 0 \Rightarrow x+1=0 \Rightarrow x=-1$$ So, there's only one value of x responsible for making the denominator zero, which is x = -1. 3.

Identify the vertical asymptote

Since x = -1 makes the denominator zero, there is a vertical asymptote at this value: $$x = -1$$ In conclusion, the given function: $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$ has one vertical asymptote located at x = -1.

Key concepts

Analytical Methods in Calculus

Analytical methods in calculus are techniques used to solve calculus problems without numerical approximation. These include algebraic manipulation, differentiation, integration, and limits. When tackling rational functions, such as the given exercise \(h(x)=\frac{e^{x}}{(x+1)^{3}}\), we use an analytical method by finding points where the function cannot exist, like vertical asymptotes.

A vertical asymptote occurs when the function's denominator approaches zero, and its value reaches an infinitely large absolute value. To find these asymptotes analytically, we set the denominator to zero and solve for the variable, \(x\). This method requires strong algebraic skills and understanding of limits, as we determine where the function behaves erratically or is undefined.

In essence, analytical methods help us comprehensively evaluate the behavior of functions, giving precise results without relying on approximations or graphics.

Graphing Utility

A graphing utility, such as a graphing calculator or computer software, aids in visualizing a function's behavior by plotting its graph. When it comes to identifying vertical asymptotes, a graphing utility can be particularly helpful as it provides a visual representation where the function's graph approaches but does not touch a vertical line.

For the exercise involving the function \(h(x)=\frac{e^{x}}{(x+1)^{3}}\), a graphing utility can confirm the analytical findings. By entering the function into the utility and carefully analyzing the graph, students can see the behavior around \(x = -1\) and notice that the graph indeed approaches infinity near this point. This graphical approach complements the analytical approach and offers a powerful way to understand the concept.

Overall, using a graphing utility enhances comprehension by providing a visual context—particularly useful for those who are more visually inclined. The utility can visually confirm the limits and asymptotes that analytical methods determine.

Solving Rational Functions

Solving rational functions, such as \(h(x)=\frac{e^{x}}{(x+1)^{3}}\), involves finding where the function is undefined or reaches an extreme value. Rational functions are the ratio of two polynomials, and vertical asymptotes are frequently found where the denominator is zero but the numerator is not.

To solve these, we follow a series of steps. Firstly, set the denominator equal to zero and solve for \(x\). This identifies potential vertical asymptotes. Secondly, verify that the numerator does not also equal zero at these points, which would indicate a hole in the graph rather than an asymptote.

In the function provided, after setting \( (x+1)^{3}=0 \) and solving for \(x\), we find that \(x=-1\). There is no need to check the numerator, \(e^{x}\), as it never equals zero. Thus, \(x=-1\) is where the vertical asymptote exists. Such methods are an invaluable part of calculus, enabling students to analyze and understand the intricate behavior of rational functions.