Answer
See below.
Work Step by Step
The limit may be $\displaystyle \lim_{x\rightarrow a}$ or $\displaystyle \lim_{x\rightarrow\pm\infty}$
$1.$ Finding $\displaystyle \lim_{x\rightarrow a}f(x)$ algebraically, we have two strategies, one for closed forms, and one for piecewise forms.
$1.A \ \ Closed\ forms$
1.A.1 Try substituting $x=\mathrm{a}$ in the formula for $f(x)$.
Then one of the following three things may happen:
1.A.1.1 $f(\mathrm{a})$ is defined. Then $\displaystyle \lim_{x\rightarrow \mathrm{a}}f(x)=f(\mathrm{a})$.
1.A.1.2 $f(\mathrm{a})$ is not defined and has the indeterminate form $0/0$.
Try to simplify the expression for $f$ to cancel one of the terms that gives $0.$
1.A.1.3 $f(\mathrm{a})$ is not defined and has one of the determinate forms listed above in the table of determinate forms.
Use the table to determine the limit.
$1.B\ \ Piecewise form.$
Find the left and right limits at the values of $x$ where the function changes from one formula to another.
If they exist and are equal, the limit exists. Otherwise, it doesn't.
$2.$ Finding $\displaystyle \lim_{x\rightarrow\pm\infty}f(x)$ algebraically:
2.1. If the function is a polynomial or a ratio of polynomials, use Theorem 10.2
2.2 Apply determinate forms $ k^{+\infty}=\infty$ and $k^{-\infty}=0$ (if $k$ is positive).
2.3 Later in the book, we learn how to apply L'Hopital's rule for indeterminate forms.
DISADVANTAGES:
Sometimes, this method (algebraic determination of limits) can be very difficult, when the function is a complicated combination of functions.
For example,
$\displaystyle \lim_{x\rightarrow\infty}(e^{x}-x^{2}\ln x),\qquad$
which has the form
$\infty-\infty.$
