Question

Slant asymptote The linear function \(\ell(x)=m x+b,\) for finite \(m \neq 0,\) is a slant asymptote of \(f(x)\) if \(\lim _{x \rightarrow \infty}(f(x)-\ell(x))=0\) a. Use a graphing utility to make a sketch that shows \(\ell(x)=x\) is a slant asymptote of \(f(x)=x\) tanh \(x .\) Does \(f\) have any other slant asymptotes? b. Provide an intuitive argument showing that \(f(x)=x \tanh x\) behaves like \(\ell(x)=x\) as \(x\) gets large. c. Prove that \(\ell(x)=x\) is a slant asymptote of \(f\) by confirming \(\lim _{x \rightarrow \infty}(x \tanh x-x)=0\)

Short answer

If so, prove it. Answer: Yes, the function \(f(x)=x\tanh{x}\) has a slant asymptote \(\ell(x)=x\). This can be proved by showing that \(\lim_{x\to\infty}(x\tanh{x}-x)=0\).

Step-by-step solution

Sketching the graph

Utilize a graphing utility to plot both functions, \(f(x)=x\tanh{x}\) and \(\ell(x)=x\). Based on the graph, we can observe that \(\ell(x)=x\) appears to be a suitable slant asymptote for \(f(x)\). There doesn't seem to be any other slant asymptotes.

Intuitive argument

To provide an intuitive argument, recall the definition of the hyperbolic tangent function: \(\tanh{x}=\frac{\sinh{x}}{\cosh{x}}\). Let's examine the behavior of the \(\tanh{x}\) function for large values of \(x\): As \(x\to\infty\), \(\cosh{x}\to\infty\). Thus, as \(x\) gets larger, \(\tanh{x}\) approaches 1. Hence, \(x\tanh{x}\) behaves like \(x\) times a number very close to 1, which means that it approximates the linear function \(\ell(x)=x\) as \(x\) becomes large.

Proving the slant asymptote

In order to confirm that \(\ell(x)=x\) (with m=1 and b=0) is a slant asymptote of the \(f(x)=x\tanh{x}\), we need to verify that \(\lim_{x\to\infty}(x\tanh{x}-x)=0\). To do this, let's first rewrite the limit expression: \(\lim_{x\to\infty}(x\tanh{x}-x)=\lim_{x\to\infty}(x\tanh{x}-x\times1)=\lim_{x\to\infty}(x(\tanh{x}-1))\) Now, as \(x\to\infty, \tanh{x}\to{1}\). Hence, we obtain: \(\lim_{x\to\infty}(x(\tanh{x}-1))=\lim_{x\to\infty}(x(1-1))=\lim_{x\to\infty}(x(0))=0\) Therefore, \(\ell(x)=x\) is indeed a slant asymptote of \(f(x)=x\tanh{x}\).

Key concepts

Hyperbolic Functions

Hyperbolic functions, such as the hyperbolic tangent (\tanh{}), are analogs of the ordinary trigonometric functions but for a hyperbola instead of a circle.

They are defined using exponential functions and have properties that look somewhat similar to trigonometric functions, yet they bear distinct behavior. The hyperbolic sine and cosine, denoted by \(\sinh{x}\) and \(\cosh{x}\), respectively, are the foundational functions from which others like \(\tanh{x}\) are derived.

For instance, \(\tanh{x}\) is calculated as the ratio of \(\sinh{x}\) over \(\cosh{x}\): \[\tanh{x} = \frac{\sinh{x}}{\cosh{x}}\]. As \(x\) becomes large, the values of \(\tanh{x}\) approach 1, which is crucial in understanding the behavior of functions with \(\tanh{x}\) when discussing asymptotic behavior and limits.

Limits in Calculus

Limits in calculus are a fundamental concept that deal with the behavior of functions as they approach a specific point or infinity.

The limit of a function \(f(x)\) as \(x\) approaches a value \(c\) is symbolized as \(\lim_{x \to c}f(x)\) and represents the value that \(f(x)\) gets closer to as \(x\) nears \(c\). If \(x\) approaches infinity, we look at the behavior of \(f(x)\) as it grows without bound.

In the context of the provided exercise, we examine \(\lim _{x \rightarrow \infty}(x \tanh x-x)\) to determine the presence of a slant asymptote. This involves understanding how \(x \tanh x\) behaves relative to the linear function \(\ell(x)=x\) as \(x\) increases indefinitely.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves at the extreme ends of its domain, often as the input approaches infinity or negative infinity.

A slant (or oblique) asymptote is a line that a graph of a function approaches but never touches as \(x\) moves towards infinity or negative infinity. It is different from horizontal asymptotes because the slant asymptote can have a nonzero slope. The presence of a slant asymptote like \(\ell(x)=mx+b\) suggests that the function grows similar to this line at its extremities.

The confirmation of \(\ell(x)=x\) as a slant asymptote to the function \(f(x)=x\tanh{x}\) rests on showing that the difference between the functions, \(f(x)-\ell(x)\), approaches zero as \(x\) tends towards infinity, illustrated by the limit \(\lim _{x \rightarrow \infty}(f(x)-\ell(x))=0\). This means that the farther out we go along the x-axis, the closer the curve of \(f(x)\) gets to the line \(\ell(x)\), embodying the concept of a slant asymptote.