determine-all-vertical-asymptotes-xaxa-of-the-function-fx2x3x48x316x2fleftxright

Question

Determine all vertical asymptotes x=ax=a of the function f(x)=2x+3x4−8x3+16x2f\left(xight)=\frac{2x+3}{$$x^4-8x^3$+16x^2$$}. Evaluate lim⁡x→a+f(x){\displaystyle\lim_{x	o $$a^{+}$$}}f\left(xight), lim⁡x→a−f(x){\displaystyle\lim_{x	o a_{}^{-}}}f\left(xight), and lim⁡x→a f(x){\displaystyle\lim_{x	o a}}	ext{ }f\left(xight) for each value of aa.

Accepted Answer

﻿x=0x=0x=0﻿; ﻿lim⁡x→0+f(x){\displaystyle\lim_{x\$$to0^{+}$$}}f\left(xight)x→0+lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→0−f(x){\displaystyle\lim_{x\$$to0^{-}$$}}f\left(xight)x→0−lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→0f(x){\displaystyle\$$lim_{x	o0}f(x$$)}x→0lim​f(x)﻿ ﻿=∞=\infty=∞﻿﻿x=4x=4x=4﻿; ﻿lim⁡x→4+f(x){\displaystyle\lim_{x\$$to4^{+}$$}}f\left(xight)x→4+lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→4−f(x){\displaystyle\lim_{x\$$to4^{-}$$}}f\left(xight)x→4−lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→4f(x)\displaystyle \lim_{x 	o 4} f(x)x→4lim​f(x)﻿ ﻿=∞=\infty=∞﻿

Answer

﻿x=0x=0x=0﻿; ﻿lim⁡x→0+f(x){\displaystyle\lim_{x\$$to0^{+}$$}}f\left(xight)x→0+lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→0−f(x){\displaystyle\lim_{x\$$to0^{-}$$}}f\left(xight)x→0−lim​f(x)﻿ ﻿===﻿ −∞-\infty; ﻿lim⁡x→0f(x){\displaystyle\$$lim_{x	o0}f(x$$)}x→0lim​f(x)﻿ does not exist	ext{does not exist}﻿x=4x=4x=4﻿; ﻿lim⁡x→4+f(x){\displaystyle\lim_{x\$$to4^{+}$$}}f\left(xight)x→4+lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→4−f(x){\displaystyle\lim_{x\$$to4^{-}$$}}f\left(xight)x→4−lim​f(x)﻿ ﻿===﻿ −∞-\infty; ﻿lim⁡x→4f(x)\displaystyle \lim_{x 	o 4} f(x)x→4lim​f(x)﻿ does not exist	ext{does not exist}

Answer

﻿x=0x=0x=0﻿; ﻿lim⁡x→0+f(x){\displaystyle\lim_{x\$$to0^{+}$$}}f\left(xight)x→0+lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→0−f(x){\displaystyle\lim_{x\$$to0^{-}$$}}f\left(xight)x→0−lim​f(x)﻿ ﻿===﻿ ﻿−∞-\infty−∞﻿; ﻿lim⁡x→0f(x){\displaystyle\$$lim_{x	o0}f(x$$)}x→0lim​f(x)﻿ ﻿does not exist	ext{does not exist}does not exist﻿

Answer

﻿x=4x=4x=4﻿; ﻿lim⁡x→4+f(x){\displaystyle\lim_{x\$$to4^{+}$$}}f\left(xight)x→4+lim​f(x)﻿ ﻿===﻿ ﻿∞\infty∞﻿; ﻿lim⁡x→4−f(x){\displaystyle\lim_{x\$$to4^{-}$$}}f\left(xight)x→4−lim​f(x)﻿ ﻿===﻿ ﻿−∞-\infty−∞﻿; ﻿lim⁡x→4f(x)\displaystyle \lim_{x 	o 4} f(x)x→4lim​f(x)﻿ ﻿does not exist	ext{does not exist}does not exist﻿