Limits: Algebraically

Given the graphs of \(f\) and \(g\), find the following limits.

1. \({\displaystyle \lim_{x\to -6} \ [f(x)+g(x)]}\)
2. \({\displaystyle \lim_{x\to  5} \ [f(x)-g(x)]}\)
3. \({\displaystyle \lim_{x\to 5} \ 3g(x)}\)
4. \({\displaystyle \lim_{x\to -3} \ [f(x)g(x)]}\)
5. \({\displaystyle \lim_{x\to -6} \ \dfrac{f(x)}{g(x)}}\)
6. \({\displaystyle \lim_{x\to 5} \ [f(x)]^3}\)
7. \({\displaystyle \lim_{x\to 5} \ \sqrt[3]{g(x)}}\)
8. \({\displaystyle \lim_{x\to 5} \ \dfrac{f(x)-g(x)}{3g(x)}}\)

Use the Properties of Limits to find the following limits algebraically, and verify your answer using the Direct Substitution Property.

1. \(\displaystyle \lim_{x\to 7} \ (5x-x^2)\)
2. \(\displaystyle \lim_{x\to -4} \dfrac{x^3-2}{2x^2}\)

Given the function \(g\) below, find the following limits algebraically, if they exist.
\[
g(x) =
\begin{cases}
x^2-4& \text{ \(x\leq0\)} \\
-4 & \text{ \(x>0\)} \\
\end{cases}
\]

1. \({\displaystyle \lim_{x\to 0^-} g(x)}\)
2. \({\displaystyle \lim_{x\to 0^+} g(x)}\)
3. \({\displaystyle \lim_{x\to 0} g(x)}\)
4. \({\displaystyle \lim_{x\to 3} g(x)}\)
5. \({\displaystyle \lim_{x\to -3} g(x)}\)

Find \(\displaystyle \lim_{x\to 3}\ \dfrac{x^2-2x-3}{3x+4}\) algebraically. If the limit does not exist, state so and use limit notation to describe any infinite behavior.

Find \(\displaystyle \lim_{x\to 3}\ \dfrac{x^2-3x}{2x^2-5x-3}\) algebraically. If the limit does not exist, state so and use limit notation to describe any infinite behavior.

Find \(\displaystyle \lim_{x\to -3} \dfrac{\dfrac{1}{x+2}+1}{x+3}\) algebraically. If the limit does not exist, state so and use limit notation to describe any infinite behavior.

Find \(\displaystyle \lim_{x\to -1} \dfrac{\sqrt{x+2}-1}{x+1}\) algebraically. If the limit does not exist, state so and use limit notation to describe any infinite behavior.

Find \(\displaystyle \lim_{x\to -7} \dfrac{x+7}{|x+7|}\) algebraically. If the limit does not exist, state so and use limit notation to describe any infinite behavior.