Mathematics · Ch 7 — Applications of Differential Calculus

Asymptotes

7.9.2

Asymptotes

An asymptote for the curve $y=f(x)$ is a straight line which is, informally, "a tangent at infinity" to the curve — the distance between the curve and the line tends to $0$ as the point on the curve runs off to infinity. There are three types.

1. Horizontal asymptote (parallel to the $x$ -axis). $y=L$ is a horizontal asymptote for $y=f(x)$ if either

\lim_{x\to+\infty}f(x)=L\qquad\text{or}\qquad\lim_{x\to-\infty}f(x)=L.

2. Vertical asymptote (parallel to the $y$ -axis). $x=a$ is a vertical asymptote for $y=f(x)$ if

\lim_{x\to a^{-}}f(x)=\pm\infty\qquad\text{or}\qquad\lim_{x\to a^{+}}f(x)=\pm\infty.

3. Slant (oblique) asymptote. Occurs when, for a rational function, the numerator's degree is exactly one higher than the denominator's. Found by dividing the numerator by the denominator (long or synthetic division): the quotient (a linear polynomial) is the slant asymptote; the remainder over the denominator vanishes as $x\to\pm\infty$ .

Worked patterns (Examples 7.66–7.68):

$f(x)=\dfrac1x$ : as $x\to0^-$ , $f\to-\infty$ ; as $x\to0^+$ , $f\to+\infty$ — so $x=0$ is a vertical asymptote; by symmetry (the curve is unchanged under $x\leftrightarrow y$ ), $y=0$ is a horizontal asymptote too, giving the familiar rectangular-hyperbola shape.
$\dfrac{x^2-6x+7}{x+5}$ -type expression: numerator degree (2) exceeds denominator degree (1) by exactly one $\Rightarrow$ a slant asymptote exists; long division gives the linear quotient directly as the asymptote's equation (the curve approaches this line but never actually touches it). …

Figure 7.29Graph of the line y = x, symmetric with respect to the origin.

Fig. 7.29 — Graph of the line y = x, symmetric with respect to the origin.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Graph of the line y = x, symmetric with respect to the origin. …

Figure 7.30Graph of the parabola y = x^2, symmetric with respect to the y-axis.

Fig. 7.30 — Graph of the parabola y = x^2, symmetric with respect to the y-axis.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Graph of the parabola y = x^2, symmetric with respect to the y-axis. …

Figure 7.31Graph of the sideways parabola x = y^2, opening to the right and symmetric with respect to the x-axis.

Fig. 7.31 — Graph of the sideways parabola x = y^2, opening to the right and symmetric with respect to the x-axis.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Graph of the sideways parabola x = y^2, opening to the right and symmetric with respect to the x …

Figure 7.32Graph of y = 1/x, a rectangular hyperbola with the y-axis (x=0) as vertical asymptote and the x-axis (y=0) as horizontal asymptote.

Fig. 7.32 — Graph of y = 1/x, a rectangular hyperbola with the y-axis (x=0) as vertical asymptote and the x-axis (y=0) as horizontal asymptote.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Graph of y = 1/x, a rectangular hyperbola with the y-axis (x=0) as vertical asymptote and the x-axis (y=0) as horizontal …

Figure 7.33Graph of f(x) = (x^2 - 6x + 7)/(x + 5) with its slant (oblique) asymptote y = x - 11; the curve approaches the line but never crosses it. (Axes compressed by a factor of 10.)

Fig. 7.33 — Graph of f(x) = (x^2 - 6x + 7)/(x + 5) with its slant (oblique) asymptote y = x - 11; the curve approaches the line but never crosses it. (Axes compressed by a factor of 10.)

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Graph of f(x) = (x^2 - 6x + 7)/(x + 5) with its slant (oblique) asymptote y = x - 11; the curve approaches the line but never crosses it. (Axes compressed by …

Figure 7.34Graph of f(x) = (2x^2 - 8)/(x^2 - 16) with vertical asymptotes x = -4 and x = 4 and horizontal asymptote y = 2; the central branch peaks at (0, 1/2).

Fig. 7.34 — Graph of f(x) = (2x^2 - 8)/(x^2 - 16) with vertical asymptotes x = -4 and x = 4 and horizontal asymptote y = 2; the central branch peaks at (0, 1/2).

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Graph of f(x) = (2x^2 - 8)/(x^2 - 16) with vertical asymptotes x = -4 and x = 4 and horizontal asymptote y = 2; the central branch peaks …

Figure 7.35Curve sketch of y = x^2 - x - 6 = (x-3)(x+2): x-intercepts (-2,0) and (3,0), y-intercept (0,-6), vertex (local minimum) at (1/2, -25/4).

Fig. 7.35 — Curve sketch of y = x^2 - x - 6 = (x-3)(x+2): x-intercepts (-2,0) and (3,0), y-intercept (0,-6), vertex (local minimum) at (1/2, -25/4).

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Curve sketch of y = x^2 - x - 6 = (x-3)(x+2): x-intercepts (-2,0) and (3,0), y-intercept (0,-6), vertex (local minimum) at (1 …

Figure 7.36Curve sketch of y = x^3 - 6x - 9: x-intercept (3,0), y-intercept (0,-9), local maximum at (-sqrt2, 4sqrt2 - 9) and local minimum at (sqrt2, -4sqrt2 - 9), point of inflection at (0,-9).

Fig. 7.36 — Curve sketch of y = x^3 - 6x - 9: x-intercept (3,0), y-intercept (0,-9), local maximum at (-sqrt2, 4sqrt2 - 9) and local minimum at (sqrt2, -4sqrt2 - 9), point of inflection at (0,-9).

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Curve sketch of y = x^3 - 6x - 9: x-intercept (3,0), y-intercept (0,-9), local maximum at (-sqrt2, 4sqrt2 - 9) and local minimum at (sqrt2, -4sqrt2 - 9), point of inf …

Figure 7.37Curve sketch of y = (x^2 - 3x)/(x - 1) = x(x-3)/(x-1): passes through the origin and (3,0), with vertical asymptote x = 1.

Fig. 7.37 — Curve sketch of y = (x^2 - 3x)/(x - 1) = x(x-3)/(x-1): passes through the origin and (3,0), with vertical asymptote x = 1.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Curve sketch of y = (x^2 - 3x)/(x - 1) = x(x-3)/(x-1): passes through the origin and (3,0), with vertical asympto …

Figure 7.38Curve sketch of y = 3x/(x^2 - 1): vertical asymptotes x = -1 and x = 1, horizontal asymptote y = 0, point of inflection at the origin (0,0).

Fig. 7.38 — Curve sketch of y = 3x/(x^2 - 1): vertical asymptotes x = -1 and x = 1, horizontal asymptote y = 0, point of inflection at the origin (0,0).

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What this figure shows. Curve sketch of y = 3x/(x^2 - 1): vertical asymptotes x = -1 and x = 1, horizontal asymptote y = 0, point of inflection at the or …

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