Dr. Xiaozhou Li

Dr. Xiaozhou Li

Mathematics. Strength. Long-Term Thinking. Think Hard. Train Harder. Live Long.

Problem Sets: Functions, Limits (函数、极限)

Published:

Last Update: 2026-09-28

Remark:

  • In accordance with the course sequence (late-transcendentals), this problem set does not yet include problems involving exponential functions, inverse functions, or logarithms. These topics will be covered in subsequent problem sets.
  • This problem set focuses on limits. Do not use L’Hôpital’s Rule or Taylor expansions; they are not needed here.
  • The Additional and Advanced Exercises are somewhat more involved than the standard exercises. They may require multi-step reasoning, conceptual analysis, modest proof components, or a combination of several ideas.

Functions

  • Let $f(x) = \sqrt{3 + 2x - x^{2}}$, find its domain and range.

  • Let $f(x) = \sqrt{1-x}$ and $g(x) = \dfrac{x+1}{x-2}$, find $(f\circ g)(x)$ and its domain.

  • Let $f(x)=\begin{cases} 1, & \lvert x \rvert \leq 1 \\ 0, & \lvert x \rvert > 1\end{cases} $, find $(f\circ f) (x)$.

  • Let $f(x)=\begin{cases} \sin x, & x<1 \\ x - 1, & x\geq 1 \end{cases} $, $g(x)=\begin{cases} x+2, & x<0 \\ x^{2} - 1, & x\geq 0 \end{cases}$, find $(f\circ g)(x)$.

Limits

The $\varepsilon - \delta$ definition

  • Let $f(x)=\dfrac{1}{x+2}$.
    1. Show that if $\lvert x-2\rvert<1$, then $\left\lvert f(x)-\dfrac14\right\rvert<\dfrac{\lvert x-2\rvert}{12}$.
    2. Find $\delta>0$ such that if $\lvert x-2\rvert<\delta$, then $\left\lvert f(x)-\dfrac14\right\rvert <0.01$.
    3. Prove rigorously that $\displaystyle\lim_{x\to 2} f(x)=\frac14$.

  • Prove directly from the definition that $\displaystyle \lim _{x \to -1} (3x+ 4) = 1$.

  • Prove directly from the definition that $\displaystyle \lim _{x \to 1} (x^{2} -x) = 6$.

  • Prove directly from the definition that $\displaystyle \lim_{x\to 9}\sqrt{x-5}=2$.

  • Uniqueness of the Limit. Prove that a function converges to at most one limiting value. In other words, use the limit definition to prove that if $\lim\limits_{x\to c} f (x) = L_1$ and $\lim\limits_{x\to c} f (x) =L_2$, then $L_1 = L_2$.

  • Prove the following Limit Laws using the definition of the limit:
    • Sum Rule
    • Product Rule
    • Quotient Rule.

  • Prove the Squeeze Theorem using the definition of the limit.

  • Let $f(x)=1$ if $x$ is rational and $f(x)=0$ if $x$ is irrational. Prove that $\lim_{x\to c} f(x)$ does not exist for any $c$. Hint: There exist rational and irrational numbers arbitrarily close to any $c$.

Find the following limits (if they exist)

  • $\displaystyle \lim _{x \to 0^{+}} \dfrac{\sqrt{\cos x}-1}{\tan \sqrt[3]{x}}$

  • $\displaystyle\lim _{x \to 0^{+}} \dfrac{\cos \sqrt{x}-1}{\sqrt{\tan x}}$

  • $\displaystyle\lim _{x \to 0} \dfrac{\sqrt{1 + x^{2}} - 1}{3 \sin x}$

  • $\displaystyle\lim _{x \to 1} \dfrac{\sqrt{2x}(x-1)}{\lvert x-1 \rvert}$

  • $\displaystyle\lim _{x \to 0} \cos\dfrac{1}{x}$

  • $\displaystyle\lim _{x \to 0} x\cos\dfrac{1}{x}$

  • $\displaystyle\lim _{x \to 0} \dfrac{1 - \cos m x}{ \sin 2x}\,(m\neq 0)$

  • $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos x}{ \tan x}$

  • $\displaystyle\lim_{x \to 0} \dfrac{\tan x - \sin x}{x^{3}}$

  • $\displaystyle\lim _{x \to \frac{\pi}{2}}\left(\frac{1}{\tan x -1} - \frac{2}{\tan^2 x -1}\right)$

  • $\displaystyle \lim _{x \to 0}f(x)$, where $\displaystyle f(x)=\begin{cases} x^{2} \sin (1 / x), & x<0 \\ \sqrt{x}, & x>0 \end{cases}$ .

  • $\displaystyle \lim_{x\to 0}\frac{3\sin^2 x+x^3\cos\frac{1}{x}}{(1+\cos x)\tan^2 x}$.

  • $\displaystyle\lim_{x\to 0^{+}} \dfrac{\frac{1}{x} + \sin\dfrac{1}{x} + 3\frac{1}{\sqrt{x}}}{\dfrac{1}{x} + \sin\dfrac{1}{x}}$

  • $\displaystyle\lim_{x\to 1} \left(\dfrac{1}{1-x} - \dfrac{3}{1 - x^{3}}\right)$

  • $\displaystyle\lim _{x \to 1} \dfrac{1-x^{2}}{\sin \pi x}$

  • $\displaystyle\lim _{x \to 0} \dfrac{\cos \sqrt{x}-1}{\sqrt[3]{x+1}-1}$

  • $\displaystyle \lim_{x\to 0} \lfloor x\rfloor$

  • $\displaystyle \lim_{x\to 0} \dfrac{\lfloor x\rfloor}{x}$

  • $\displaystyle \lim_{x\to 0} x \left\lfloor \dfrac{1}{x}\right\rfloor$

  • $\displaystyle \lim_{x\to \infty}\dfrac{4 - 3x^{3}}{\sqrt{x^{6} + 9}}$

  • $\displaystyle\lim_{x\to -\infty}\dfrac{4 - 3x^{3}}{\sqrt{x^{6} + 9}}$

  • $\displaystyle\lim_{x\to \infty}\dfrac{3x^{2}+5}{5x+3}\sin\frac{2}{x}$

  • $\displaystyle \lim_{x\to-\infty}\left(1+2x+\sqrt{4x^2+3x-2}\right)$

  • $\displaystyle \lim_{x\to \infty}\dfrac{x^{2}+x}{\lfloor{x}\rfloor}$

  • $\displaystyle \lim _{x \to \infty} \dfrac{x^{2}+x}{\lceil x\rceil^{2}}$

  • $\displaystyle \lim _{x \to \infty} \dfrac{\sin x}{\lfloor x\rfloor}$

  • If $\displaystyle\lim_{x\to a}[f(x)+g(x)] = 2$ and $\displaystyle\lim_{x\to a}[f(x)-g(x)] = 1$, find $\displaystyle\lim_{x\to a}[f(x)g(x)]$.

  • If $\displaystyle \lim_{x\to 2}\frac{f(x)-1}{x-2}=2026$, find $\displaystyle \lim_{x\to 2}f(x)$.

  • Find the oblique asymptote of $f(x) = \dfrac{x^{2} + 2x + 3}{x+1}$.

  • Find all vertical, horizontal, and oblique asymptotes of $f(x) = \dfrac{x^{2}+1}{x-1}$.

  • Find the constants $a$ and $b$, such that $\displaystyle \lim_{x \to \infty} \left( \sqrt{ax^{2} - x + 3} - 2x \right) = b$.

  • Given that $\displaystyle \lim_{x \to \infty} (3x - \sqrt{ax^{2} + bx + 1}) = 1$, find the values of $a$ and $b$.

  • Find the values of $a$ and $b$ such that $\displaystyle\lim_{x \to \infty} \left( \dfrac{x^{2} + 1}{x + 1} - ax - b \right) = 0$.

  • Using a diagram of the unit circle and the Pythagorean Theorem, show that \(\sin^2\theta \le (1-\cos\theta)^2+\sin^2\theta \le \theta^2\) Conclude that $\sin^2\theta \le 2(1-\cos\theta) \le \theta^2$ and use this to give an alternative proof that the limit in Exercise 51 equals $0$. Then give an alternative proof of the result in Exercise 54.

  1. (a) Investigate the limit $\displaystyle\lim_{x\rightarrow c}\frac{\sin x-\sin c}{x-c}$ numerically for the five values $c=0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}$. (b) Can you guess the answer for general $c$? (c) Check numerically that your answer to (b) works for two other values of $c$.

Additional and Advanced Exercises

  • $\displaystyle \lim_{x \to 0} \dfrac{1 - \cos x\sqrt{\cos 2x}\sqrt{\cos 3x}}{\sin x\tan x}$

  • $\displaystyle \lim_{x\to 0^{+}} \dfrac{1 - \sqrt{\cos x}}{x(1-\cos\sqrt{x})}$

  • $\displaystyle \lim_{x \to 0} \dfrac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^{3}}$

  • $\displaystyle \lim_{x\to 0}\frac{3\sin^2 x+x^3\cos\frac{1}{x}}{(1+\cos x)\tan^2 x}$

  • $\displaystyle \lim_{x\to 0}\frac{1 - \sqrt{1-x^2}\cos x}{1+x^2 - \cos^2 x}$

  • $\displaystyle \lim_{x \to \infty} \left( \sqrt[3]{x^{3} + 3x} - \sqrt{x^{2} - 2x} \right)$

  • $\displaystyle \lim_{x \to \infty} \left( \sin\sqrt{x+1} - \sin\sqrt{x}\right)$

  • Find the constants $a, b$ and $c$ such that $\displaystyle\lim_{x\to 1} \dfrac{a(x-1)^{2} + b(x-1) + c - \sqrt{x^{2}+3}}{(x-1)^{2}} =0$.

  • Find all oblique asymptotes of $f(x) = \sqrt{\dfrac{x^{3}}{x-1}}$.