Question

Find all vertical asymptotes ﻿x=ax=ax=a﻿ of the following functions. For each value of ﻿aaa﻿, determine ﻿lim⁡x→a+f(x){\displaystyle\lim_{x	o $$a^{+}$$}}f\left(xight)x→a+lim​f(x)﻿, ﻿lim⁡x→a−f(x){\displaystyle\lim_{x	o $$a^{-}$$}}f\left(xight)x→a−lim​f(x)﻿, and ﻿lim⁡x→af(x){\displaystyle\lim_{x	o a}}f\left(xight)x→alim​f(x)﻿.f(x)=x+1x3−4x2+4xf\left(xight)=\frac{x+1}{$$x^3$-4x^2+4x$$}

Accepted Answer

Identify the points where the denominator of the function f(x) = \frac{x+1}{$$x^3$-4x^2+4x$$} is equal to zero, as these are potential vertical asymptotes. Set $$x^3$$ - $$4x^2 + 4x = 0 $$and solve for x. Factor the equation $$x^3$$ - $$4x^2 + 4x = 0. $$Start by factoring out the greatest common factor, which is x, to get $$x(x^2 - 4x + 4) = 0. $$Further factor the quadratic $$x^2 - 4x + 4. $$Notice that it is a perfect square trinomial, so it can be factored as (x - $$2)^2. $$Thus, the equation becomes x(x - $$2)^2 = 0. $$Solve the factored equation x(x - $$2)^2 = 0 to $$find the values of x that make the denominator zero. The solutions are x = 0 and x = 2. For each potential vertical asymptote x = 0 and x = 2, evaluate the one-sided limits: \lim_{x 	o a^+} f(x), \lim_{x 	o a^-} f(x), and \lim_{x 	o a} f(x). Analyze the behavior of the function as x approaches these values from the left and right to confirm the presence of vertical asymptotes.

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Key Concepts

Vertical Asymptotes

Limits

Rational Functions