Homework 3

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. A function $g(x)$ has the following graph: Find the following. If they do not exist explain why.
    1. $\displaystyle\lim_{x\rightarrow -3} g(x)$
    2. $\displaystyle\lim_{x\rightarrow 2^-} g(x)$
    3. $\displaystyle\lim_{x\rightarrow 2^+} g(x)$
    4. $\displaystyle\lim_{x\rightarrow 2} g(x)$
    5. $\displaystyle\lim_{x\rightarrow -1} g(x)$
    6. $g(3)$
  2. Given the following functions $f(x)$ and $g(x)$: Find the following limits, if it exists. If it doesn't, explain why.
    1. $\displaystyle \lim_{x\rightarrow 2}\ [f(x) + g(x)]$
    2. $\displaystyle \lim_{x\rightarrow 1}\ [f(x) + g(x)]$
    3. $\displaystyle \lim_{x\rightarrow 0}\ [f(x)g(x)]$
    4. $\displaystyle \lim_{x\rightarrow 2}\ [x^3f(x)]$
    5. $\displaystyle \lim_{x\rightarrow 1}\ \sqrt{3 + f(x)}$
  3. Suppose $f(x)$ and $g(x)$ are two different functions. You compute the limit: \[\lim_{x\rightarrow 5} \dfrac{f(x)}{g(x)} = \cdots = \dfrac{0}{0}\]
    1. What type of result is this called?
    2. Suppose you decide to leave your answer as $\dfrac{0}{0}$. Is your answer correct and complete?
    3. What will always happen which allows you to remove the $0$'s from the numerator and denominator?
  4. Find the following limits using limit laws. If it is an indeterminate form of type $\dfrac{0}{0}$, make sure to properly deal with it.
    1. $\displaystyle\lim_{x\rightarrow 1} \sqrt{\dfrac{2x^2 + 2}{2x^2 - 1}}$
    2. $\displaystyle \lim_{x\rightarrow 5}\dfrac{x^2 - 6x + 5}{x - 5}$
    3. $\displaystyle \lim_{h\rightarrow 0}\dfrac{(4+h)^2 - 16}{h}$
    4. $\displaystyle \lim_{x\rightarrow 0}\dfrac{\sqrt{x^2 + 9} - 3}{x^2}$
    5. $\displaystyle \lim_{x\rightarrow 16}\dfrac{4 - \sqrt{x}}{16x - x^2}$
    6. $\displaystyle \lim_{x\rightarrow 0}\dfrac{1}{x} - \dfrac{1}{x^2 + x}$
    7. $\displaystyle \lim_{t\rightarrow-3}\dfrac{t^2 - 9}{2t^2 + 7t +3}$
    8. $\displaystyle \lim_{h\rightarrow 0} \dfrac{\dfrac{1}{3 +h} - \dfrac{1}{3}}{h}$
  5. For the following problems, sketch the graph of $f$ and find $\displaystyle \lim_{x\rightarrow a} f(x)$, if it exists.
    1. $f(x) = \begin{cases} x^2 & x \leq 1 \\ x-1 & x > 1\end{cases}, \qquad a = 1$
    2. $f(x) = \begin{cases} x^2 - 3 & x \neq 0 \\ 0 & x = 0\end{cases}, \qquad a = 0$
    3. $f(x) = \begin{cases} -1 & x < 0 \\ x - 1 & x > 0 \end{cases}, \qquad a = 0$.
  6. Use the squeeze theorem to show \[\lim_{x\rightarrow 0}x^2 \cos (20\pi x) = 0\]
  7. If $ 4x - 9 \leq f(x) \leq x^2 - 4x + 7$ for all $x \geq 0$, find $\displaystyle\lim_{x\rightarrow 4}f(x)$.
  8. 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. $\displaystyle\lim_{x\rightarrow 3^-} f(x)= \lim_{x\rightarrow 3^+}f(x)$
    7. (skip this one) $f(x)$ is continuous from the right at $x = 1$.
  9. Suppose a function $f(x)$ has $f(2) = 5$ and $\displaystyle \lim_{x \rightarrow 2}f(x) = 5$. Is the function continuous at $x = 2$?
  10. Your professor asks you to use the definition of continuity to show $f(x)$ is continuous at $x=1$. Write down the three conditions you must verify in order for your answer to be considered correct.

    11. The rest of these problems will appear on next week's homework. Skip the rest of these for Homework 3.

  11. Use the definition of continuity to explain why the function is continuous at the given number $a$.
    1. $\displaystyle f(x) = \begin{cases}(x-1)^2& x < 0 \\ (x+1)^2 & x \geq 0\end{cases} \qquad a = 0$
    2. $\displaystyle f(x) = \begin{cases}x(x-1)& x < 1 \\ 0 & x = 1 \\ \sqrt{x-1} & x > 1\end{cases} \qquad a = 1$
  12. Use the definition of continuity to explain why the function is discontinuous at the given number $a$. Sketch the graph of the function.
    1. $\displaystyle f(x) = \begin{cases}x+1 & x < 0 \\ x^2 & x \geq 0\end{cases} \qquad a = 0$
    2. $\displaystyle f(x) = \begin{cases}x + 5 & x < 0 \\ 2 & x = 0 \\ -x^2 + 5 & x > 0\end{cases} \qquad a = 0$
  13. If a function $f(x)$ is continuous at $x = a$, what does $\displaystyle \lim_{x\rightarrow a} f(x)$ have to be?
  14. How would you define $f(2)$ in the function \[f(x) = \dfrac{x^2 - x - 2}{x - 2}\] in order to make $f(x)$ continuous at $x = 2$?
  15. Where is the function \[f(x) = \dfrac{\sin(x)}{x + 1}\] continuous?
  16. What's the difference between a removeable, jump and infinite discontinuity?
  17. Draw one graph of a function which satisfies the following conditions simultaneously:
    • Jump discontinuity at $2$ but continuous from the right at $2$
    • Discontinuous at $-1$ and $4$ but continuous from the left at $-1$ and from the right at $4$.
    • Continuous everywhere else