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Find any vertical and horizontal asymptotes, if any: f(x)=\dfrac{5x}{\sqrt{x^2+4}}

Question:

Find any vertical and horizontal asymptotes, if any: {eq}f(x)=\dfrac{5x}{\sqrt{x^2+4}} {/eq}

Asymptotes of Functions:

An asymptote of a function is a line which the graph of that function gets arbitrarily close to. More specifically:

1) A function {eq}f(x) {/eq} will have the line {eq}x=a {/eq} as a vertical asymptote if either {eq}\lim_{x \to a^-}=\pm \infty {/eq} or {eq}\lim_{x \to a^+}=\pm \infty {/eq}.

2) A function {eq}f(x) {/eq} will have the line {eq}y=b {/eq} as a horizontal asymptote if {eq}\lim_{x \to \infty} f(x) = b {/eq} or {eq}\lim_{x \to -\infty} f(x) = b {/eq}.

Answer and Explanation: 1

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We're examining the function {eq}f(x)=\dfrac{5x}{\sqrt{x^2+4}} {/eq}. Note that

{eq}\begin{align*} x^2+4&\ge 0^2+4\\ &=4 \end{align*} {/eq}

and...

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Finding Asymptotes Using Limits

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Chapter 12 / Lesson 7
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A function can have a vertical asymptote, a horizontal asymptote and more generally, an asymptote along any given line (e.g., y = x). In this lesson, we learn how to find all asymptotes by evaluating the limits of a function.


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