Homework 5

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. Suppose \[\lim_{x\rightarrow 1^-} f(x) = 3 \qquad \lim_{x\rightarrow 1^+} f(x) = 3.01\] Does $\displaystyle\lim_{x\rightarrow 1}f(x)$ exist? If not, why?
  2. Given this function $f(x)$ find the following:
    1. $\displaystyle\lim_{x\rightarrow 2^-}f(x)$
    2. $\displaystyle\lim_{x\rightarrow 2^+}f(x)$
    3. $\displaystyle\lim_{x\rightarrow 2}f(x)$
    4. $\displaystyle\lim_{x\rightarrow 4^-}f(x)$
    5. $\displaystyle\lim_{x\rightarrow 4^+}f(x)$
    6. $\displaystyle\lim_{x\rightarrow 4}f(x)$
    7. $f(2)$
    8. $f(4)$
  3. Given this function $f(x)$ find the following:
    1. $\displaystyle\lim_{x\rightarrow -3^-}f(x)$
    2. $\displaystyle\lim_{x\rightarrow -3^+}f(x)$
    3. $\displaystyle\lim_{x\rightarrow -3}f(x)$
    4. $\displaystyle\lim_{x\rightarrow 0^-}f(x)$
    5. $\displaystyle\lim_{x\rightarrow 0^+}f(x)$
    6. $\displaystyle\lim_{x\rightarrow 0}f(x)$
    7. $f(0)$
    8. $\displaystyle\lim_{x\rightarrow 5^-}f(x)$
    9. $\displaystyle\lim_{x\rightarrow 5^+}f(x)$
  4. Given this function $f(x)$ find the following:
    1. $\displaystyle\lim_{x\rightarrow 0}f(x)$
    2. $\displaystyle\lim_{x\rightarrow 2^-}f(x)$
    3. $\displaystyle\lim_{x\rightarrow 2^+}f(x)$
    4. $\displaystyle\lim_{x\rightarrow \infty}f(x)$
    5. $\displaystyle\lim_{x\rightarrow -\infty}f(x)$
    6. Find all vertical and horizontal asymptotes.
  5. Use a table or graph to find the following limits:
    1. $\displaystyle\lim_{x\rightarrow 3^+}\ln(x^2 - 9)$
    2. $\displaystyle\lim_{x\rightarrow 5^-}\dfrac{e^x}{(x-5)^3}$
  6. 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. $g(3)$
  7. Using the table or graph method, determine if the following limits are $-\infty$ or $\infty$:
    1. $\displaystyle \lim_{x\rightarrow-3^+}\dfrac{x+2}{x+3}$
    2. $\displaystyle \lim_{x\rightarrow-3^-}\dfrac{x+2}{x+3}$
    3. $\displaystyle \lim_{x\rightarrow1}\dfrac{2-x}{(x-1)^2}$
  8. 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. $\displaystyle \lim_{x\rightarrow 1}\ \sqrt{3 + f(x)}$
  9. Find the following limits using limit laws:
    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. $\displaystyle \lim_{x\rightarrow 0}\dfrac{1}{x} - \dfrac{1}{x^2 + x}$
      Hint: subtract the two fractions first.
    7. $\displaystyle \lim_{t\rightarrow-3}\dfrac{t^2 - 9}{2t^2 + 7t +3}$
    8. $\displaystyle \lim_{h\rightarrow 0}\dfrac{\dfrac{1}{3 + h} - \dfrac{1}{3}}{h}$
    9. (skip this) $\displaystyle \lim_{x\rightarrow 0}\dfrac{\sin 3x}{x}$
    10. (skip this) $\displaystyle \lim_{t\rightarrow 0}\dfrac{\tan 6t}{\sin 2t}$
    11. (skip this) $\displaystyle \lim_{x\rightarrow 0}\dfrac{\sin^2 4t}{t^2}$
  10. 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. $f(x) = \begin{cases} x^2 - 3 & x \neq 0 \\ 0 & x = 0\end{cases}, \qquad a = 0$
  11. 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. $\displaystyle\lim_{x\rightarrow 3^-} f(x)= \lim_{x\rightarrow 3^+}f(x)$

    12. The rest of these problems will appear on next week's homework. Skip the rest for this homework.

  12. Use the squeeze theorem to show \[\lim_{x\rightarrow 0}x^2 \cos (20\pi x) = 0\]
  13. If $ 4x - 9 \leq f(x) \leq x^2 - 4x + 7$ for all $x \geq 0$, find $\lim_{x\rightarrow 4}f(x)$.
  14. Suppose a function $f(x)$ has $f(2) = 5$ and $\displaystyle \lim_{x \rightarrow 2}f(x) = 5$. Is the function continuous at $x = 2$?
  15. Use the definition of continuity to explain why the function is discontinuous at the given number $a$. Sketch the graph of the function.
    1. $\displaystyle f(x) = \begin{cases}e^x & x < 0 \\ x^2 & x \geq 0\end{cases} \qquad a = 0$
    2. $\displaystyle f(x) = \begin{cases}x + 5 & x < 0 \\ 2 & x = 0 \\ -x^2 + 5 & x > 0\end{cases} \qquad a = 0$