Calculus

Limit Theory

1. 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

2. 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 $\varepsilon=1$.

3. 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 $x=a$.

4. 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?

5. Which of the following is a properly set up difference quotient for the derivative of $f(x)=2^x$ at $x=4$?
• $\lim\limits_{x \to \infty}{\frac{2^{4+h}-16}{h}}$
• $\lim\limits_{x \to \infty}{\frac{2^{4+h}}{h}}$
• $\lim\limits_{x \to 0}{\frac{16(2^h-1)}{h}}$
• $\lim\limits_{x \to 0}{\frac{2^{4+h}}{h}}$

Limits at Points
1. Find $\lim\limits_{x \to 0}{f(x)}$.

1. Find $\lim\limits_{x \to 0}{\frac{x}{\left|x\right|}}$.

2. Evaluate $\lim\limits_{x \to 0}{\frac{x^2-\sin(x)}{\cos(x)}}$.

3. 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 $\lim\limits_{x \to 0}{f'(x)}$?

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. $\lim\limits_{x \to 1}{f'(x)}=\frac{a}{b}$, where $a$ and $b$ are both positive integers and $\frac{a}{b}$ is in reduced form. what is $ab$?

5. $\lim\limits_{x \to 4}{\frac{\sqrt x + 2}{x-4}}=$

6. $\lim\limits_{x \to 0}{\sqrt{x^3-x^2}}=$

7. Given $f(x)=x^3+x+1$ and $g(x)=f^{-1}(x)$, evaluate $\lim\limits_{h \to 0}{\frac{g(1+h)-g(1)}{h}}$.

8. Let $f(x)=\log_{9}(x^2+4)$. Find $\lim\limits_{h \to 0}{\frac{f(x+h)-f(x-h)}{h}}$.

9. 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 $x=2$?

Limits Approaching Infinity
1. $\lim\limits_{x \to \infty}{\frac{6x^4-7x^5+x^7+x-13}{8x^5-17x^6+3x^2-3x^3-2x^7+1}}=$

2. Evaluate: $\lim\limits_{x \to -\infty}{\frac{10-2^x}{10+2^{-x}}}$

3. Find $\lim\limits_{x \to \infty}{\frac{e^x}{x\cdot2^x}}$

4. Evaluate: $\lim\limits_{n \to \infty}{\frac{6^n+2^n}{8^n}}$

5. Find the value of $\lim\limits_{x \to \infty}{x^{2/x}}$

Lâ€™Hospitalâ€™s Limits
1. $\lim\limits_{h \to 0}{\frac{\tan^{-1}(1+h)-\frac{\pi}{4}}{h}}=$

2. $\lim\limits_{x \to 3}{\frac{3x^3-7x^2-7x+3}{2x^3-13x^2+26x-15}}=$

3. If $f(x)=\frac{x^2+5x-24}{x^2+10x+16}$, then $\lim\limits_{x \to -8}{f(x)}=$

4. Evaluate the following limit: $\lim\limits_{x \to 3}{\frac{x^2-x-6}{x-(2x+3)^{\frac{1}{2}}}}$

5. Lily the Ladybug loves limits. She kindly asks you to find: $\lim\limits_{x \to 0}{\frac{(x+\pi)^3+6(x+\pi)^2-\pi^3-6\pi^2}{x}}$.

6. What is $\lim\limits_{x \to 0}{\frac{2\ln(1+x)}{x}}$?

7. Find the value of $\lim\limits_{x \to 2^-}{\frac{\left|x-2\right|}{x-2}}$

8. Find $\lim\limits_{x \to 0}{\frac{\ln(\frac{x}{k}+1)}{x}}$

9. Compute the limit: $\lim\limits_{x \to 0}{\frac{sin^2(2x)}{x^2}}$

Limits Involving $e$
1. $\lim\limits_{x \to \infty}{\left(1-\frac{3}{2x}\right)^{\frac{4}{5}}}=$

2. Find $\lim\limits_{n \to \infty}{\left(1+\frac{1}{2n}\right)^{4n}}$

3. Find the value of $\lim\limits_{x \to \infty}{\left(1+\frac{1}{10^x}\right)^{10^x}}$

4. Evaluate the limit: $\lim\limits_{x \to -\infty}{\left(\frac{-9+x}{x}\right)^{-x/3}}$

5. Find $\lim\limits_{x \to \infty}{\left(\frac{x}{x+3}\right)^{-8x}}$

6. $\lim\limits_{x \to 0^+}{(1 +2x)^{-1/x}}=$