Other Asymptotes

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A few days ago, a reader asked if you could find the location of horizontal asymptotes from the derivative of a function. The answer, alas, is no. You can determine that a function has a horizontal asymptote from its derivative, but not where it is. This is because a function and its many possible vertical translations have the same derivative but different horizontal asymptotes. To locate it precisely, we need more information, an initial condition (a point on the curve).

I have written a post about the relationship between vertical asymptotes and the derivative of the function here. Today’s post will discuss horizontal asymptotes and their derivatives.

Definition: a straight line associated with a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero and the slope of the curve at the point approaches the slope of the line. (Merriam-Webster dictionary). Thus, asymptotes can be vertical, horizontal, or slanted.

Horizontal Asymptotes

Horizontal asymptotes are a form of end behavior: they appear as \displaystyle x\to \infty or \displaystyle x\to -\infty . Specifically, if a function has a horizontal asymptote(s), then \displaystyle \underset{{x\to \pm \infty }}{\mathop{{\lim }}}\,{f}'\left( x \right)=0,  At an asymptote, the curve must approach a horizontal line, therefore its slope must be approaching zero as \displaystyle x\to \infty or \displaystyle x\to -\infty .  The converse is not true: If the derivative approaches zero, the function may not have a horizontal asymptote. A counterexample is \displaystyle f\left( x \right)=\sqrt[3]{x}.

In the first four examples that follow, the derivative approaches 0 as \displaystyle x\to \infty and/or \displaystyle x\to -\infty . 

Example 1: \displaystyle {f}'\left( x \right)=\frac{1}{{1+{{x}^{2}}}}  (Figure 1 in blue).This is the derivative of \displaystyle f\left( x \right)={{\tan }^{{-1}}}(x),\ -\tfrac{\pi }{2}<f\left( x \right)<\tfrac{\pi }{2}. (Figure 1 in red). We see that the derivative approaches zero in both directions, this tells us that there may be horizontal asymptote(s). We must look at the function to find where they are: \displaystyle y=\tfrac{\pi }{2} and \displaystyle y=-\tfrac{\pi }{2}

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6 thoughts on “Other Asymptotes”

  1. Pingback: Reading the Derivative’s Graph | Teaching Calculus

  2. Pingback: Unit 5 – Analytical Applications of Differentiation | Teaching Calculus

  3. You wrote that if the limit of the derivative is zero, then there is a horizontal asymptote (at the relevant end). I’m not sure that’s true. Certainly the converse is true: if there is a horizontal asymptote, the derivative will approach zero. But for f(x)=cbrt(x), the derivative will approach zero at both ends, but f(x) does not have a horizontal asymptote. Or am I misunderstanding the second sentence of your fourth paragraph?

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  4. I think there might be a typo in the formula given for figure 4. Perhaps it should be 1.1^-x * sin(x) [missing the negative sign], or perhaps something like 0.9^x * sin(x)

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