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. Draw a graph of a sequence which satisfies \[\lim_{n\rightarrow \infty} a_n = 3\]
  2. Below is a sequence $a_t$ of the world record times for the men's 100 meter spring every five years.
    • From a biological perspective, do you think $\lim_{t\rightarrow\infty} a_t = 0$ or greater than zero? Justify your answer.
  3. Determine whether the following sequences are convergent or divergent. For the ones that converge, find out what it converges to.
    • $a_n = \dfrac{3 + 5n}{2 + 7n}$
    • $a_n = 2^{-n} + 6^{-n}$
    • $a_n = \dfrac{10^n}{1 + 9^n}$
    • $a_n = \dfrac{3^{n+2}}{5^n}$
    • $a_n = \dfrac{2n^3 + n - 1}{n^2 + 1}$
  4. Find the limit:
    • $\displaystyle\lim_{x\rightarrow\infty} \dfrac{3x - 2}{2x + 1}$
    • $\displaystyle\lim_{x\rightarrow\infty} \dfrac{5}{10^x}$
    • $\displaystyle\lim_{x\rightarrow\infty} \left(\sqrt{9x^2 + x} - 3x\right)$
    • $\displaystyle\lim_{x\rightarrow-\infty} (x^4 + x^5)$
    • $\displaystyle\lim_{x\rightarrow-\infty} \dfrac{1 - e^x}{1+2e^x}$
    • $\displaystyle\lim_{x\rightarrow-\infty} \left(e^{-x} + 2\cos 2x\right)$
  5. 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?
  6. 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. $f(4)$
  7. 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. $\displaystyle\lim_{x\rightarrow 5^+}f(x)$
  8. 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.
  9. Use a table or graph to find the following limits:
    1. $\displaystyle\lim_{x\rightarrow 3^+}\ln(x^2 - 9)$
    2. $\displaystyle\lim_{x\rightarrow 5^-}\dfrac{e^x}{(x-5)^3}$