limx→2x2+5x−3=−9\Large \lim\limits_{x \to 2}{\frac{x^2+5}{x-3}}=-9x→2limx−3x2+5=−9
limx→3x2−3x2+1=0\Large \lim\limits_{x \to \sqrt{3}}{\frac{x^2-3}{x^2+1}}=0x→3limx2+1x2−3=0
limx→1x2−2x+1x2−1=0\Large \lim\limits_{x \to 1}{\frac{x^2-2x+1}{x^2-1}}=0x→1limx2−1x2−2x+1=0
limx→04x3−2x2+x3x2+2x=12\Large \lim\limits_{x \to 0}{\frac{4x^3-2x^2+x}{3x^2+2x}}=\frac{1}{2}x→0lim3x2+2x4x3−2x2+x=21
limh→0(x+h)2−x2h=2x\Large \lim\limits_{h \to 0}{\frac{(x+h)^2-x^2}{h}}=2xh→0limh(x+h)2−x2=2x
limx→∞(2−1x+1x2)=2\Large \lim\limits_{x \to \infty}{(2-\frac{1}{x}+\frac{1}{x^2})}=2x→∞lim(2−x1+x21)=2
limx→∞x2−12x2−x−1\Large \lim\limits_{x \to \infty}{\frac{x^2-1}{2x^2-x-1}}x→∞lim2x2−x−1x2−1
limx→∞x2+xx4−3x2+1\Large \lim\limits_{x \to \infty}{\frac{x^2+x}{x^4-3x^2+1}}x→∞limx4−3x2+1x2+x
limx→4x2−6x+8x2−5x+4\Large \lim\limits_{x \to 4}{\frac{x^2-6x+8}{x^2-5x+4}}x→4limx2−5x+4x2−6x+8
limx→∞(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)
limx→∞(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)
limx→∞1+2+3...+(n−1)n2\Large \lim\limits_{x \to \infty}{\frac{1+2+3...+(n-1)}{n^2}}x→∞limn21+2+3...+(n−1)
limx→∞(n+1)(n+2)(n+3)5n3\Large \lim\limits_{x \to \infty}{\frac{(n+1)(n+2)(n+3)}{5n^3}}x→∞lim5n3(n+1)(n+2)(n+3)
limx→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)
limx→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
limx→∞x22x+1\Large \lim\limits_{x \to \infty}{\frac{x^2}{2x+1}}x→∞lim2x+1x2
limx→∞(2x3−x+1)\Large \lim\limits_{x \to \infty}{(2x^3-x+1)}x→∞lim(2x3−x+1)
limx→0x2sin1x\Large \lim\limits_{x \to 0}{x^2\sin{\frac{1}{x}}}x→0limx2sinx1
limx→∞arctanxx\Large \lim\limits_{x \to \infty}{\frac{\arctan x}{x}}x→∞limxarctanx
limx→∞2x+1x\Large \lim\limits_{x \to \infty}{\frac{2x+1}{x}}x→∞limx2x+1
limx→01−x21−x\Large \lim\limits_{x \to 0}{\frac{1-x^2}{1-x}}x→0lim1−x1−x2
limx→0sinωxx\Large \lim\limits_{x \to 0}{\frac{\sin{\omega x}}{x}}x→0limxsinωx
limx→0tan3xx\Large \lim\limits_{x \to 0}{\frac{\tan 3x}{x}}x→0limxtan3x
limx→0sin2xsin5x\Large \lim\limits_{x \to 0}{\frac{\sin 2x}{\sin 5x}}x→0limsin5xsin2x
limx→0xcotx\Large \lim\limits_{x \to 0}{x\cot x}x→0limxcotx
limx→01−cos2xxsinx\Large \lim\limits_{x \to 0}{\frac{1-\cos 2x}{x\sin x}}x→0limxsinx1−cos2x
limx→∞2nsinx2n\Large \lim\limits_{x \to \infty}{2^n\sin \frac{x}{2^n}}x→∞lim2nsin2nx,x 为不等于0的常数
limx→0(1−x)1x\Large \lim\limits_{x \to 0}{(1-x)^{\frac{1}{x}}}x→0lim(1−x)x1
limx→0(1+2x)1x\Large \lim\limits_{x \to 0}{(1+2x)^{\frac{1}{x}}}x→0lim(1+2x)x1
limx→∞(1+xx)2x\Large \lim\limits_{x \to \infty}{(\frac{1+x}{x})^{2x}}x→∞lim(x1+x)2x
limx→∞(1−1x)kx\Large \lim\limits_{x \to \infty}{(1-\frac{1}{x})^{kx}}x→∞lim(1−x1)kx,k 为正整数
limx→0tan3x2x\Large \lim\limits_{x \to 0}{\frac{\tan 3x}{2x}}x→0lim2xtan3x
limx→0sinxnsinmx\Large \lim\limits_{x \to 0}{\frac{\sin x^n}{\sin^m x}}x→0limsinmxsinxn,m、n 为正整数
limx→0tanx−sinxsin3x\Large \lim\limits_{x \to 0}{\frac{\tan x -\sin x}{\sin^3x}}x→0limsin3xtanx−sinx
limx→0sinx−tanx(1+x23−1)(1+sinx−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
limx→0x2−2x+5\Large \lim\limits_{x \to 0}{\sqrt{x^2-2x+5}}x→0limx2−2x+5
limα→π4sin32α\Large \lim\limits_{\alpha \to \frac{\pi}{4}}{\sin^3 2\alpha}α→4πlimsin32α
limx→π6ln(2cos2x)\Large \lim\limits_{x \to \frac{\pi}{6}}{\ln(2\cos 2x)}x→6πlimln(2cos2x)
limx→0x+1−1x\Large \lim\limits_{x \to 0}{\frac{\sqrt{x+1}-1}{x}}x→0limxx+1−1
limx→15x−4−xx−1\Large \lim\limits_{x \to 1}{\frac{\sqrt{5x-4}-\sqrt{x}}{x-1}}x→1limx−15x−4−x
limx→αsinx−sinαx−α\Large \lim\limits_{x \to \alpha}{\frac{\sin x-\sin \alpha}{x-\alpha}}x→αlimx−αsinx−sinα
limx→+∞(x2+x−x2−x)\Large \lim\limits_{x \to +\infty}{(\sqrt{x^2+x}-\sqrt{x^2-x})}x→+∞lim(x2+x−x2−x)
limx→∞e1x\Large \lim\limits_{x \to \infty}{e^{\frac{1}{x}}}x→∞limex1
limx→0lnsinxx\Large \lim\limits_{x \to 0}{\ln{\frac{\sin x}{x}}}x→0limlnxsinx
limx→∞(1+1x)x2\Large \lim\limits_{x \to \infty}{(1+\frac{1}{x})^{\frac{x}{2}}}x→∞lim(1+x1)2x
limx→0(1+3tan2x)cot2x\Large \lim\limits_{x \to 0}{(1+3\tan^2x)^{\cot^2x}}x→0lim(1+3tan2x)cot2x
limx→∞(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
limx→01+tanx−1+sinxx1+sin2x−x\Large \lim\limits_{x \to 0}{\frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x\sqrt{1+\sin^2x}-x}}x→0limx1+sin2x−x1+tanx−1+sinx
limx→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
limx→+∞x(x2+1−x)\Large \lim\limits_{x \to +\infty}{x(\sqrt{x^2+1}-x)}x→+∞limx(x2+1−x)
limx→∞(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
limx→0tanx−sinxx3\Large \lim\limits_{x \to 0}{\frac{\tan x-\sin x}{x^3}}x→0limx3tanx−sinx
limx→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
limx→π2(sinx)tanx\Large \lim\limits_{x \to \frac{\pi}{2}}{(\sin x)^{\tan x}}x→2πlim(sinx)tanx