lim⁡x→2x2+5x−3=−9\Large \lim\limits_{x \to 2}{\frac{x^2+5}{x-3}}=-9x→2lim​x−3x2+5​=−9

lim⁡x→3x2−3x2+1=0\Large \lim\limits_{x \to \sqrt{3}}{\frac{x^2-3}{x^2+1}}=0x→3​lim​x2+1x2−3​=0

lim⁡x→1x2−2x+1x2−1=0\Large \lim\limits_{x \to 1}{\frac{x^2-2x+1}{x^2-1}}=0x→1lim​x2−1x2−2x+1​=0

lim⁡x→04x3−2x2+x3x2+2x=12\Large \lim\limits_{x \to 0}{\frac{4x^3-2x^2+x}{3x^2+2x}}=\frac{1}{2}x→0lim​3x2+2x4x3−2x2+x​=21​

lim⁡h→0(x+h)2−x2h=2x\Large \lim\limits_{h \to 0}{\frac{(x+h)^2-x^2}{h}}=2xh→0lim​h(x+h)2−x2​=2x

lim⁡x→∞(2−1x+1x2)=2\Large \lim\limits_{x \to \infty}{(2-\frac{1}{x}+\frac{1}{x^2})}=2x→∞lim​(2−x1​+x21​)=2

lim⁡x→∞x2−12x2−x−1\Large \lim\limits_{x \to \infty}{\frac{x^2-1}{2x^2-x-1}}x→∞lim​2x2−x−1x2−1​

lim⁡x→∞x2+xx4−3x2+1\Large \lim\limits_{x \to \infty}{\frac{x^2+x}{x^4-3x^2+1}}x→∞lim​x4−3x2+1x2+x​

lim⁡x→4x2−6x+8x2−5x+4\Large \lim\limits_{x \to 4}{\frac{x^2-6x+8}{x^2-5x+4}}x→4lim​x2−5x+4x2−6x+8​

lim⁡x→∞(1+1x)(2−1x2)\Large \lim\limits_{x \to \infty}{(1+\frac{1}{x})(2-\frac{1}{x^2})}x→∞lim​(1+x1​)(2−x21​)

lim⁡x→∞(1+12+14+...+12n)\Large \lim\limits_{x \to \infty}{(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^n})}x→∞lim​(1+21​+41​+...+2n1​)

lim⁡x→∞1+2+3...+(n−1)n2\Large \lim\limits_{x \to \infty}{\frac{1+2+3...+(n-1)}{n^2}}x→∞lim​n21+2+3...+(n−1)​

lim⁡x→∞(n+1)(n+2)(n+3)5n3\Large \lim\limits_{x \to \infty}{\frac{(n+1)(n+2)(n+3)}{5n^3}}x→∞lim​5n3(n+1)(n+2)(n+3)​

lim⁡x→1(11−x−31−x3)\Large \lim\limits_{x \to 1}{(\frac{1}{1-x}-\frac{3}{1-x^3})}x→1lim​(1−x1​−1−x33​)

lim⁡x→2x3+2x2(x−2)2\Large \lim\limits_{x \to 2}{\frac{x^3+2x^2}{(x-2)^2}}x→2lim​(x−2)2x3+2x2​

lim⁡x→∞x22x+1\Large \lim\limits_{x \to \infty}{\frac{x^2}{2x+1}}x→∞lim​2x+1x2​

lim⁡x→∞(2x3−x+1)\Large \lim\limits_{x \to \infty}{(2x^3-x+1)}x→∞lim​(2x3−x+1)

lim⁡x→0x2sin⁡1x\Large \lim\limits_{x \to 0}{x^2\sin{\frac{1}{x}}}x→0lim​x2sinx1​

lim⁡x→∞arctan⁡xx\Large \lim\limits_{x \to \infty}{\frac{\arctan x}{x}}x→∞lim​xarctanx​

lim⁡x→∞2x+1x\Large \lim\limits_{x \to \infty}{\frac{2x+1}{x}}x→∞lim​x2x+1​

lim⁡x→01−x21−x\Large \lim\limits_{x \to 0}{\frac{1-x^2}{1-x}}x→0lim​1−x1−x2​

lim⁡x→0sin⁡ωxx\Large \lim\limits_{x \to 0}{\frac{\sin{\omega x}}{x}}x→0lim​xsinωx​

lim⁡x→0tan⁡3xx\Large \lim\limits_{x \to 0}{\frac{\tan 3x}{x}}x→0lim​xtan3x​

lim⁡x→0sin⁡2xsin⁡5x\Large \lim\limits_{x \to 0}{\frac{\sin 2x}{\sin 5x}}x→0lim​sin5xsin2x​

lim⁡x→0xcot⁡x\Large \lim\limits_{x \to 0}{x\cot x}x→0lim​xcotx

lim⁡x→01−cos⁡2xxsin⁡x\Large \lim\limits_{x \to 0}{\frac{1-\cos 2x}{x\sin x}}x→0lim​xsinx1−cos2x​

lim⁡x→∞2nsin⁡x2n\Large \lim\limits_{x \to \infty}{2^n\sin \frac{x}{2^n}}x→∞lim​2nsin2nx​，x 为不等于0的常数

lim⁡x→0(1−x)1x\Large \lim\limits_{x \to 0}{(1-x)^{\frac{1}{x}}}x→0lim​(1−x)x1​

lim⁡x→0(1+2x)1x\Large \lim\limits_{x \to 0}{(1+2x)^{\frac{1}{x}}}x→0lim​(1+2x)x1​

lim⁡x→∞(1+xx)2x\Large \lim\limits_{x \to \infty}{(\frac{1+x}{x})^{2x}}x→∞lim​(x1+x​)2x

lim⁡x→∞(1−1x)kx\Large \lim\limits_{x \to \infty}{(1-\frac{1}{x})^{kx}}x→∞lim​(1−x1​)kx，k 为正整数

lim⁡x→0tan⁡3x2x\Large \lim\limits_{x \to 0}{\frac{\tan 3x}{2x}}x→0lim​2xtan3x​

lim⁡x→0sin⁡xnsin⁡mx\Large \lim\limits_{x \to 0}{\frac{\sin x^n}{\sin^m x}}x→0lim​sinmxsinxn​，m、n 为正整数

lim⁡x→0tan⁡x−sin⁡xsin⁡3x\Large \lim\limits_{x \to 0}{\frac{\tan x -\sin x}{\sin^3x}}x→0lim​sin3xtanx−sinx​

lim⁡x→0sin⁡x−tan⁡x(1+x23−1)(1+sin⁡x−1)\Large \lim\limits_{x \to 0}{\frac{\sin x-\tan x}{(\sqrt[3]{1+x^2}-1)(\sqrt{1+\sin x}-1)}}x→0lim​(31+x2​−1)(1+sinx​−1)sinx−tanx​

lim⁡x→0x2−2x+5\Large \lim\limits_{x \to 0}{\sqrt{x^2-2x+5}}x→0lim​x2−2x+5​

lim⁡α→π4sin⁡32α\Large \lim\limits_{\alpha \to \frac{\pi}{4}}{\sin^3 2\alpha}α→4π​lim​sin32α

lim⁡x→π6ln⁡(2cos⁡2x)\Large \lim\limits_{x \to \frac{\pi}{6}}{\ln(2\cos 2x)}x→6π​lim​ln(2cos2x)

lim⁡x→0x+1−1x\Large \lim\limits_{x \to 0}{\frac{\sqrt{x+1}-1}{x}}x→0lim​xx+1​−1​

lim⁡x→15x−4−xx−1\Large \lim\limits_{x \to 1}{\frac{\sqrt{5x-4}-\sqrt{x}}{x-1}}x→1lim​x−15x−4​−x​​

lim⁡x→αsin⁡x−sin⁡αx−α\Large \lim\limits_{x \to \alpha}{\frac{\sin x-\sin \alpha}{x-\alpha}}x→αlim​x−αsinx−sinα​

lim⁡x→+∞(x2+x−x2−x)\Large \lim\limits_{x \to +\infty}{(\sqrt{x^2+x}-\sqrt{x^2-x})}x→+∞lim​(x2+x​−x2−x​)

lim⁡x→∞e1x\Large \lim\limits_{x \to \infty}{e^{\frac{1}{x}}}x→∞lim​ex1​

lim⁡x→0ln⁡sin⁡xx\Large \lim\limits_{x \to 0}{\ln{\frac{\sin x}{x}}}x→0lim​lnxsinx​

lim⁡x→∞(1+1x)x2\Large \lim\limits_{x \to \infty}{(1+\frac{1}{x})^{\frac{x}{2}}}x→∞lim​(1+x1​)2x​

lim⁡x→0(1+3tan⁡2x)cot⁡2x\Large \lim\limits_{x \to 0}{(1+3\tan^2x)^{\cot^2x}}x→0lim​(1+3tan2x)cot2x

lim⁡x→∞(3+x6+x)x−12\Large \lim\limits_{x \to \infty}{(\frac{3+x}{6+x})^{\frac{x-1}{2}}}x→∞lim​(6+x3+x​)2x−1​

lim⁡x→01+tan⁡x−1+sin⁡xx1+sin⁡2x−x\Large \lim\limits_{x \to 0}{\frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x\sqrt{1+\sin^2x}-x}}x→0lim​x1+sin2x​−x1+tanx​−1+sinx​​

lim⁡x→1x2−x+1(x−1)2\Large \lim\limits_{x \to 1}{\frac{x^2-x+1}{(x-1)^2}}x→1lim​(x−1)2x2−x+1​

lim⁡x→+∞x(x2+1−x)\Large \lim\limits_{x \to +\infty}{x(\sqrt{x^2+1}-x)}x→+∞lim​x(x2+1​−x)

lim⁡x→∞(2x+32x+1)x+1\Large \lim\limits_{x \to \infty}{(\frac{2x+3}{2x+1})^{x+1}}x→∞lim​(2x+12x+3​)x+1

lim⁡x→0tan⁡x−sin⁡xx3\Large \lim\limits_{x \to 0}{\frac{\tan x-\sin x}{x^3}}x→0lim​x3tanx−sinx​

lim⁡x→0(ax+bx+cx3)1x\Large \lim\limits_{x \to 0}{(\frac{a^x+b^x+c^x}{3})^{\frac{1}{x}}}x→0lim​(3ax+bx+cx​)x1​，a、b、c 都大于 0

lim⁡x→π2(sin⁡x)tan⁡x\Large \lim\limits_{x \to \frac{\pi}{2}}{(\sin x)^{\tan x}}x→2π​lim​(sinx)tanx