# Homework 4

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.

1. Determine whether the following sequences are convergent or divergent. For the ones that converge, find out what it converges to.
1. $a_n = 2^{-n} + 6^{-n}$
2. $a_n = \dfrac{10^n}{1 + 9^n}$
3. $a_n = \dfrac{3^{n+2}}{5^n}$
4. $a_n = \dfrac{n^5 + n^3}{4n^2 + \sqrt[3]{n^{19}}}$
5. $a_n = \dfrac{2^n + 3^n}{5^n}$
6. $a_n = \dfrac{e^n + 2e^n}{1 + e^n}$
7. $a_n = \dfrac{3^n}{1 + 4^n}$
8. $a_n = \dfrac{4^{-2n} - 2^{-n}}{4^n + 2^n}$
9. $a_n = \dfrac{n^3}{\sqrt{n^7 + 1}}$
2. Find the limit:
1. $\displaystyle\lim_{x\rightarrow \infty} 3^{-x}$
2. $\displaystyle\lim_{x\rightarrow\infty} \dfrac{3x - 2}{2x + 1}$
3. $\displaystyle\lim_{x\rightarrow\infty} \dfrac{5}{10^x}$
4. $\displaystyle\lim_{x\rightarrow\infty} \left(\sqrt{9x^2 + x} - 3x\right)$
5. $\displaystyle\lim_{x\rightarrow\infty} (x^4 + x^5)$
6. $\displaystyle\lim_{x\rightarrow-\infty} (x^4 + x^5)$
7. $\displaystyle\lim_{x\rightarrow-\infty} \dfrac{1 - e^x}{1+2e^x}$
8. $\displaystyle\lim_{x\rightarrow-\infty} \left(e^{-x} + 2\cos 2x\right)$
3. Suppose $f(x)$ is a function. Explain in English the intuition (not the definition) behind the symbols $\lim_{x\rightarrow a}f(x) = L$ means.
4. In class we discussed two ways of estimating the expression $\lim_{x\rightarrow a} f(x)$ Does the limit see what happens exactly at the $x$-value $a$?
5. Using the table method, find the following limits. Then compare the limit with plugging in the limit value.
1. $\displaystyle \lim_{x\rightarrow 3} \dfrac{x^2 - 3x}{x^2 - 9}$
2. $\displaystyle \lim_{t\rightarrow 0} \dfrac{\sqrt{t^2 + 4} - 2}{t^2}$
3. $\displaystyle \lim_{h\rightarrow 0} \dfrac{(h - 3)^3 + 27}{h}$
4. $\displaystyle \lim_{x\rightarrow 9} \dfrac{\sqrt{x} - 3}{x^2 - 9x}$
5. $\displaystyle \lim_{t\rightarrow 0} \dfrac{\sin t}{t^2}$