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Topos Uranos
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Asymptotes and curve sketching techniques

Summary

This lesson closes the study of limits in this unit with their first geometric application: asymptotes. In Algebra and Geometry, the vertical, horizontal and oblique asymptotes of rational functions and of functions with radicals were obtained with the sign chart and polynomial division, and the behavior of the curve near them was admitted as an intuitive fact. Here they are defined with one-sided limits, limits at infinity and infinite limits, and those admitted facts are proved. It is proved that a line is an asymptote exactly when the distance from the curve to it tends to zero, the formula for the slope and the intercept of the oblique asymptote is obtained, and it is deduced that at each end there is at most one. The lesson also corrects a common belief, that a curve cannot cross its asymptote, proposes a method for sketching the graph of a function without derivatives, and applies it to functions from Algebra and Geometry, whose sign charts and polynomial divisions are now completed with limits.

Learning objectives

  1. Define vertical, horizontal and oblique asymptotes with limits, and relate them to the distance from the curve to a line.
  2. Prove the formulas for the slope and the intercept of the oblique asymptote, and the uniqueness of the non-vertical asymptote at each end.
  3. Prove the facts admitted in Algebra and Geometry about the behavior of a rational function near a zero of its denominator, and determine the position of a curve relative to its asymptote.
  4. Sketch the graphs of algebraic and trigonometric functions by combining the domain, the symmetries, the intercepts, the sign chart and the asymptotes.

Definition

In the lesson on rational functions of Algebra and Geometry, an asymptote was a line that the curve approaches ever more closely as it moves away along one of its branches. With the limits of this unit, this idea can be written precisely. There are two ways of moving away: the variable approaches a number while the values grow without bound, or the variable grows without bound. Vertical asymptotes correspond to the first; the others, to the second.

DefinitionAsymptotes

Let ff be a function with domain AA.

  • The line x=ax = a is a vertical asymptote of the graph of ff if at least one of the one-sided limits of ff at aa is ∞\infty or −∞-\infty.
  • If AA is not bounded above, the line y=mx+by = mx + b is an asymptote of the graph of ff at ∞\infty if lim⁡x→∞(f(x)−(mx+b))=0\lim_{x \to \infty} \left(f(x) - (mx + b)\right) = 0. It is called horizontal if m=0m = 0 and oblique if m≠0m \neq 0. The asymptote at −∞-\infty is defined in the same way.

Three remarks make the definition precise. First: the horizontal asymptote y=Ly = L at ∞\infty is simply the statement lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L, from the lesson on limits as the variable tends to infinity. Second: non-vertical asymptotes are defined at each end separately, and they can be different at ∞\infty and at −∞-\infty, like the lines y=π2y = \frac{\pi}{2} and y=−π2y = -\frac{\pi}{2} of the arctangent. Third: if f(x)−g(x)f(x) - g(x) tends to 00 for a function gg that is not of the first degree, such as the quotient polynomial of a division when the degree of the numerator exceeds that of the denominator by two or more, the curve y=g(x)y = g(x) is called a curvilinear asymptote; everything that follows about oblique asymptotes holds for it, except the formulas for the slope and the intercept.

The definition measures the difference of the ordinates, that is, the vertical distance between the curve and the line. The intuitive idea, on the other hand, speaks of the distance from the curve to the line, which is measured along the perpendicular. The following proposition shows that both tend to zero together.

PropositionThe distance to the asymptote

Let Px=(x,f(x))P_x = (x, f(x)) be the point of the graph with abscissa xx, and let rr be the line y=mx+by = mx + b. Then rr is an asymptote of the graph of ff at ∞\infty if and only if lim⁡x→∞d⁡(Px,r)=0\lim_{x \to \infty} \operatorname{d}(P_x, r) = 0. The distance to the line is the vertical distance multiplied by a positive constant. In the figure, the curve is f(x)=x2+1+1x−1f(x) = \frac{x}{2} + 1 + \frac{1}{x - 1}, whose asymptote is the line y=x2+1y = \frac{x}{2} + 1.

Proof

  1. d⁡(Px,r)=∣mx−f(x)+b∣m2+1=∣f(x)−(mx+b)∣m2+1\operatorname{d}(P_x, r) = \frac{\abs{mx - f(x) + b}}{\sqrt{m^2 + 1}} = \resaltar{\frac{\abs{f(x) - (mx + b)}}{\sqrt{m^2 + 1}}}

    The line rr has the equation mx−y+b=0mx - y + b = 0. By the formula for the distance from a point to a line in the lesson on vectors in the plane, the distance from PxP_x to rr is the vertical distance divided by m2+1\sqrt{m^2 + 1}, which does not depend on xx.

  2. f(x)−(mx+b)→0⇔d⁡(Px,r)→0f(x) - (mx + b) \to 0 \Leftrightarrow \resaltar{\operatorname{d}(P_x, r) \to 0}

    By the lemma on the limit as a distance that tends to zero, from the lesson on the algebra of limits, carried over to infinity, f(x)−(mx+b)f(x) - (mx + b) tends to 00 if and only if its absolute value tends to 00; and the latter tends to 00 if and only if its product by the positive constant 1m2+1\frac{1}{\sqrt{m^2 + 1}} does, by the rule for linear combinations.