Which of the following theorems/definitions is used to show that $\lim\limits_{x \to \infty}{\frac{\cos(x)}{x}}=0$?
Central Limit Theorem
$\delta - \varepsilon$ Limit Definition
Descartes' Theorem
Squeeze Theorem
The $\varepsilon-\delta$ precise definition of a limit states that for some function $f(x)$, where $\lim\limits_{x \to a}{f(x)}=L$, there exists some values $\varepsilon$ amd $\delta$ such that $\left|f(x)-L\right|<\varepsilon$ when $0<\left|x-a\right|<\delta$. Find the maximum value of $\delta$ that can be used to prove that $\lim\limits_{x \to 2}{(10x-2)}=18$ when $\varepsilon=1$.
Which of the following statements are always true?
If $f(x)$ is continuous at $x=a$, the limit of $f(x)$ exists at $x=a$.
If $f(x)$ is differentiable at $x=a$, the limit of $f(x)$ exists at $x=a$.
If $\lim\limits_{x \to a^-}{f(x)}=\lim\limits_{x \to a^+}{f(x)}$, the limit of $f(x)$ exists at $x=a$.
If $f(x)$ is monotonic, the limit of $f(x)$ exists at $x=a$.
Mycroft Holmes is in the midst of solving a limit mystery caused by the ink washing off of an old priceless relic: A 20 year old MAO test. He knows that the limit is in the form $\lim\limits_{x \to \infty}{\frac{f(x)}{g(x)}}$, nd that $f(x)$ and $g(x)$ are distinct functions from the set $\{ \log(x^5), \log(2^x), 2x+\sin(x), x^3, 2^{x+\log(x)}\}$. Thinking in terms of his catchphrase, "balance of probability", what is the probability that the limit works out to zero?
Which of the following is a properly set up difference quotient for the derivative of $f(x)=2^x$ at $x=4$?
Let $f'(x)=\frac{x-\sqrt{x}}{x-1}$ where $x\neq 1$, and let $F(x)+C=\int f'(x)dx$ where defined. What is $\lim\limits_{x \to 0}{f'(x)}$?
Let $f'(x)=\frac{x-\sqrt{x}}{x-1}$ where $x\neq 1$, and let $F(x)+C=\int f'(x)dx$ where defined. $\lim\limits_{x \to 1}{f'(x)}=\frac{a}{b}$, where $a$ and $b$ are both positive integers and $\frac{a}{b}$ is in reduced form. what is $ab$?
$\lim\limits_{x \to 4}{\frac{\sqrt x + 2}{x-4}}=$
$\lim\limits_{x \to 0}{\sqrt{x^3-x^2}}=$
Given $f(x)=x^3+x+1$ and $g(x)=f^{-1}(x)$, evaluate $\lim\limits_{h \to 0}{\frac{g(1+h)-g(1)}{h}}$.
Let $f(x)=\log_{9}(x^2+4)$. Find $\lim\limits_{h \to 0}{\frac{f(x+h)-f(x-h)}{h}}$.
If $h(x)=x^3+2x$, what is the value of $\lim\limits_{k \to 0}{\frac{h(x+2k)-h(x-3k)}{2k}}$ at $x=2$?