Dr. Xiaozhou Li

Dr. Xiaozhou Li

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

Problem Sets: Functions, Limits (函数、极限)

Published:

Last Update: 2025-10-24

Remark:

  • Per the sequence of the course outline, this set does not yet cover problems related to exponential functions, inverse functions, and logarithms—these will be included in subsequent sets.
  • 根据课程大纲的内容顺序安排，此部分练习暂未涵盖与指数函数、反函数及对数函数相关的题目 —— 这类题目将在后续练习中纳入。

Functions

  • 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 \(f[(g(x)]\).

Limits

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

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

  • \(\lim _{x \to 0} \dfrac{\sqrt{\cos x}-1}{\tan \sqrt[3]{x}}\)

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

  • \(\lim _{x \to 0} \dfrac{1 - \cos x\sqrt{\cos 2x}\sqrt{\cos 3x}}{\sin x\tan x}\)

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

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

  • \(\lim _{x \to 0} \dfrac{1 - \cos x}{ \sin 2x}\)

  • \(\lim_{x \to 0} \dfrac{1 - \cos x}{ \tan x}\)

  • \(\lim_{x \to 0} \dfrac{\tan x - \sin x}{x^{3}}\)

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

  • \(\lim_{x\to \infty}\dfrac{x^{2}+x}{\lfloor{x}\rfloor}\)

  • \(\lim _{x \to \infty} \dfrac{x^{2}+x}{\lceil x\rceil^{2}}\)

  • \(\lim _{x \to \infty} \dfrac{\sin x}{\lfloor x\rfloor}\)

  • \(\lim _{x \to \infty} \left( \sin\sqrt{x+1} - \sin\sqrt{x}\right)\)

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

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

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

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

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

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

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

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

  • \(\lim _{x \to 1} \dfrac{1-x^{2}}{\sin \pi x}\)

  • \(\lim _{x \to 0} \dfrac{1}{x} \left(\dfrac{1}{x}-\cot x\right) \)

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

  • \(\lim _{x \to 0} \dfrac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^{3}}\)

  • \(\lim_{x \to 0} \frac{\sqrt{1 + \tan x} - \sqrt{1 + \sin x}}{x^{3}}\)

  • \(\lim_{x \to \infty} x^{2} \left(2 - x \sin \dfrac{1}{x} - \cos \dfrac{1}{x}\right)\)

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

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

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

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