# Homework 3

Directions: Show each step of your work. Turn in your answers in class on a physical piece of paper. Staple multiple sheets together. Feel free to use Desmos for graphing.

1. Given these functions find the limits. If they do not exist, explain why they do not.
1. $\displaystyle\lim_{x \rightarrow 2}[f(x) + g(x)]$
2. $\displaystyle\lim_{x \rightarrow -1}[f(x)g(x)]$
3. $\displaystyle\lim_{x \rightarrow 3}[f(-1) + g(x)]$
4. $\displaystyle\lim_{x\rightarrow 2}[x^2 \cdot f(x)]$
5. $\displaystyle\lim_{x\rightarrow 0}[f(x) + g(x)]$
6. $\displaystyle\lim_{x\rightarrow 3}\left[\dfrac{f(x)}{g(x)}\right]$
2. Find the following limits using the limit laws and fully simplify. State which laws you are using (refer to Lecture 2.4) for each equal sign.
1. $\displaystyle\lim_{x\rightarrow 1} (x + 1)$
2. $\displaystyle\lim_{t\rightarrow 0} (4t^2 + 2t + 1)$
3. $\displaystyle\lim_{x\rightarrow 1} (x + 1)$
4. $\displaystyle\lim_{x\rightarrow 3} (2x + 1)(x-1)(x+2)^2$
5. $\displaystyle\lim_{x\rightarrow 2} \sqrt{x + 2}$
6. $\displaystyle\lim_{x\rightarrow 1} \sqrt{\dfrac{2x^2 + 2}{2x^2 - 1}}$
7. $\displaystyle\lim_{x\rightarrow -1}\dfrac{x^2 - 1}{x + 1}$
8. $\displaystyle\lim_{x\rightarrow 0}\dfrac{2x^2 + x}{x}$
9. $\displaystyle\lim_{x\rightarrow -3}\dfrac{x+1}{x-3}$
3. Find the following limits with the appropriate technique.
1. $\displaystyle \lim_{x\rightarrow 1}\dfrac{\sqrt{x} - 1}{x - 1}$
2. $\displaystyle \lim_{x\rightarrow 0}\dfrac{2x^2 - 2x}{x}$
3. $\displaystyle \lim_{x\rightarrow -2}\dfrac{x^2 - 4}{x^3 + 2x^2}$
4. $\displaystyle \lim_{x\rightarrow \infty}\dfrac{x^4 + x^2 + 1}{x^3 + x + 3}$
5. $\displaystyle \lim_{x\rightarrow \infty}\dfrac{x^3 + 2x^2 - x - 6}{3x^3 -2x^2 + 4x}$
6. $\displaystyle \lim_{x\rightarrow \infty}\dfrac{x^2 + 1}{x^4 - x}$
4. It is estimated that the total population of San Luis Obispo is given by the function $P(t) = \dfrac{25t^2 + 125t + 200}{t^2 + 5t + 20}$ where $t$ is years since $2020$.
Evaluate $\displaystyle\lim_{t\rightarrow \infty} P(t)$ and interpret the meaning of the limit in English.
5. Given this graph of $f(x)$ Determine which statements are true or false.
1. $\displaystyle\lim_{x\rightarrow 0^-} f(x) = 0$
2. $\displaystyle\lim_{x\rightarrow 0^+} f(x) = 0$
3. $f(1)$ is defined.
4. $\displaystyle\lim_{x\rightarrow 2} f(x) = f(2)$
5. $\displaystyle\lim_{x\rightarrow 1} f(x)= 1$
6. $\displaystyle\lim_{x\rightarrow 3^-} f(x)= \lim_{x\rightarrow 3^+}f(x)$
6. Suppose a function $f(x)$ has $f(2) = 5$ and $\displaystyle \lim_{x \rightarrow 2}f(x) = 5$. Is the function continuous at $x = 2$?
7. Given each function, use the definition of continuity to determine where the function is discontinuous.
1. $\displaystyle f(x) = \dfrac{2}{2x - 1}$
2. $f(x) = x^{100} + 2x^{99} + 95x^5 + 99x^2 + 1$
3. $\displaystyle f(x) = \begin{cases}x & x \leq 1 \\ 2x - 1 & x > 1\end{cases}$
4. $\displaystyle f(x) = \begin{cases}x + 5 & x < 0 \\ 2 & x = 0 \\ -x^2 + 5 & x > 0\end{cases}$
8. Draw the graph of a function which is continuous on $(-\infty, -2) \cup (-2, 3) \cup (3, \infty)$ and $\displaystyle\lim_{x\rightarrow 3}f(x)$ does not exist.
9. Suppose $f(x) = 2x^{12324538742389}$. Are we allowed to use the limit laws to find $\displaystyle \lim_{x\rightarrow 3^+} f(x)$?