6973 The notion of an asymptote can be extended to includecurves as well as lines

Chapter 1, Problem 73

(choose chapter or problem)

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves \(y=f(x)\) and \(y=g(x)\) are asymptotic as \(x \rightarrow+\infty\) provided

\(\lim _{x \rightarrow+\infty}[f(x)-g(x)]=0\)

and are asymptotic as \(x \rightarrow-\infty\) provided

\(\lim _{x \rightarrow+\infty}[f(x)-g(x)]=0\)

In these exercises, determine a simpler function \(g(x)\) such that \(y=f(x)\) is asymptotic to \(y=g(x)\) as \(x \rightarrow+\infty\) or \(x \rightarrow-\infty\) Use a graphing utility to generate the graphs of \(y=f(x)\) and \(y=g(x)\) and identify all vertical asymptotes.

                                \(f(x)=\sin x+\frac{1}{x-1}\)

                                

Equation Transcription:

Text Transcription:

y=f(x)

y=g(x)

 x righarrow + infinity

lim_ x righarrow + infinity [f(x)-g(x)]=0

 x righarrow - infinity

lim_ x righarrow + infinity [f(x)-g(x)]=0

g(x)

f(x)=sin x+1/x-1