Home/Differential Calculus/Limits/Limits at infinitySection 2.6Limits at infinityFormulaCopy LaTeXlimx→∞1xp=0,p>0\lim_{x\to\infty}\frac{1}{x^p}=0,\qquad p>0x→∞limxp1=0,p>0p>0p>0p>0View detail →FormulaCopy LaTeXlimx→∞anxn+⋯+a0bmxm+⋯+b0={0,n<manbm,n=msgn (anbm)⋅∞,n>m\lim_{x\to\infty}\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}= \begin{cases} 0, & n<m\\ \frac{a_n}{b_m}, & n=m\\ \operatorname{sgn}\!\left(\frac{a_n}{b_m}\right)\cdot\infty, & n>m \end{cases}x→∞limbmxm+⋯+b0anxn+⋯+a0=⎩⎨⎧0,bman,sgn(bman)⋅∞,n<mn=mn>mBehavior of polynomial quotients as x→∞x\to\inftyx→∞ (same for x→−∞x\to-\inftyx→−∞ when degrees differ; if n>mn>mn>m, the sign depends on an/bma_n/b_man/bm and the direction).View detail →
FormulaCopy LaTeXlimx→∞1xp=0,p>0\lim_{x\to\infty}\frac{1}{x^p}=0,\qquad p>0x→∞limxp1=0,p>0p>0p>0p>0View detail →
FormulaCopy LaTeXlimx→∞anxn+⋯+a0bmxm+⋯+b0={0,n<manbm,n=msgn (anbm)⋅∞,n>m\lim_{x\to\infty}\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}= \begin{cases} 0, & n<m\\ \frac{a_n}{b_m}, & n=m\\ \operatorname{sgn}\!\left(\frac{a_n}{b_m}\right)\cdot\infty, & n>m \end{cases}x→∞limbmxm+⋯+b0anxn+⋯+a0=⎩⎨⎧0,bman,sgn(bman)⋅∞,n<mn=mn>mBehavior of polynomial quotients as x→∞x\to\inftyx→∞ (same for x→−∞x\to-\inftyx→−∞ when degrees differ; if n>mn>mn>m, the sign depends on an/bma_n/b_man/bm and the direction).View detail →