Homework 4 Directions:
Show each step of your work and fully simplify each expression.
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:
Determine whether the following sequences are convergent or divergent. For the ones that converge, find out what it converges to.
$a_n = 2^{-n} + 6^{-n}$
$a_n = \dfrac{10^n}{1 + 9^n}$
$a_n = \dfrac{3^{n+2}}{5^n}$
$a_n = \dfrac{n^5 + n^3}{4n^2 + \sqrt[3]{n^{19}}}$
$a_n = \dfrac{2^n + 3^n}{5^n}$
$a_n = \dfrac{e^n + 2e^n}{1 + e^n}$
$a_n = \dfrac{3^n}{1 + 4^n}$
$a_n = \dfrac{4^{-2n} - 2^{-n}}{4^n + 2^n}$
$a_n = \dfrac{n^3}{\sqrt{n^7 + 1}}$
Find the limit:
$\displaystyle\lim_{x\rightarrow \infty} 3^{-x}$
$\displaystyle\lim_{x\rightarrow\infty} \dfrac{3x - 2}{2x + 1}$
$\displaystyle\lim_{x\rightarrow\infty} \dfrac{5}{10^x}$
$\displaystyle\lim_{x\rightarrow\infty} \left(\sqrt{9x^2 + x} - 3x\right)$
$\displaystyle\lim_{x\rightarrow\infty} (x^4 + x^5)$
$\displaystyle\lim_{x\rightarrow-\infty} (x^4 + x^5)$
$\displaystyle\lim_{x\rightarrow-\infty} \dfrac{1 - e^x}{1+2e^x}$
$\displaystyle\lim_{x\rightarrow-\infty} \left(e^{-x} + 2\cos 2x\right)$
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.
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 $a$?
Using the table method, find the following limits. Then compare the limit with plugging in the limit value.
$\displaystyle \lim_{x\rightarrow 3} \dfrac{x^2 - 3x}{x^2 - 9}$
$\displaystyle \lim_{t\rightarrow 0} \dfrac{\sqrt{t^2 + 4} - 2}{t^2}$
$\displaystyle \lim_{h\rightarrow 0} \dfrac{(h - 3)^3 + 27}{h}$
$\displaystyle \lim_{x\rightarrow 9} \dfrac{\sqrt{x} - 3}{x^2 - 9x}$
$\displaystyle \lim_{t\rightarrow 0} \dfrac{\sin t}{t^2}$