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.

Answer the following:

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 $\displaystyle \lim_{x\rightarrow 3^+} f(x)$?