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\to $$a^{+}$$}}f\left(x\right)x→a+limf(x), lim⁡x→a−f(x){\displaystyle\lim_{x\to $$a^{-}$$}}f\left(x\right)x→a−limf(x), and lim⁡x→af(x){\displaystyle\lim_{x\to a}}f\left(x\right)x→alimf(x).f(x)=x+1x3−4x2+4xf\left(x\right)=\frac{x+1}{$$x^3$-4x^2+4x$$}

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

문제 2.4.49

2장, 문제 2.4.49

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$ , determine ${$\displaystyle\]\lim$_{x$\to$ a^{+}}}f$\left$(x$\right$)$ , ${$\displaystyle\]\lim$_{x$\to$ a^{-}}}f$\left$(x$\right$)$ , and ${$\displaystyle\]\lim$_{x$\to$ a}}f$\left$(x$\right$)$ .
$f (x) = \frac{x + 1}{x^{3} - 4 x^{2} + 4 x}$

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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 $\to$ a^+} f(x), $\lim$_{x $\to$ a^-} f(x), and $\lim$_{x $\to$ 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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주요 개념

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Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value. This typically happens when the denominator of a rational function equals zero while the numerator does not. Identifying vertical asymptotes involves finding the values of x that make the denominator zero and ensuring that the numerator is non-zero at those points.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, we evaluate the limits from both the left and right sides of the point of interest (denoted as a) to determine the behavior of the function near that point. If either limit approaches infinity, it indicates the presence of a vertical asymptote.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They are crucial in the study of vertical asymptotes because the behavior of these functions is heavily influenced by their numerator and denominator. Understanding how to factor and simplify rational functions helps in identifying points where the function may be undefined, leading to potential vertical asymptotes.