4.6 Limits at infinity and asymptotes  (Page 6/14)

Problem-solving strategy: drawing the graph of a function

Given a function $f,$ use the following steps to sketch a graph of $f\text{:}$

1. Determine the domain of the function.
2. Locate the $x$ - and $y$ -intercepts.
3. Evaluate $\underset{x\to \infty }{\text{lim}}f\left(x\right)$ and $\underset{x\to \text{−}\infty }{\text{lim}}f\left(x\right)$ to determine the end behavior. If either of these limits is a finite number $L,$ then $y=L$ is a horizontal asymptote. If either of these limits is $\infty$ or $\text{−}\infty ,$ determine whether $f$ has an oblique asymptote. If $f$ is a rational function such that $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},$ where the degree of the numerator is greater than the degree of the denominator, then $f$ can be written as
   
   $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}=g\left(x\right)+\frac{r\left(x\right)}{q\left(x\right)},$
   
   where the degree of $r\left(x\right)$ is less than the degree of $q\left(x\right).$ The values of $f\left(x\right)$ approach the values of $g\left(x\right)$ as $x\to \text{±}\infty .$ If $g\left(x\right)$ is a linear function, it is known as an oblique asymptote .
4. Determine whether $f$ has any vertical asymptotes.
5. Calculate $f^{\prime }.$ Find all critical points and determine the intervals where $f$ is increasing and where $f$ is decreasing. Determine whether $f$ has any local extrema.
6. Calculate $f\text{″}.$ Determine the intervals where $f$ is concave up and where $f$ is concave down. Use this information to determine whether $f$ has any inflection points. The second derivative can also be used as an alternate means to determine or verify that $f$ has a local extremum at a critical point.