Infinite Limits, Limits at Infinity, and Asymptotes✅

Infinite Limits, Limits at Infinity, and Asymptotes

Infinite Limits

  • Consider the behavior of the function f(x)=1x2f(x) = \frac{1}{x^2} as xx approaches 0.
  • As xx takes values closer to 0 (e.g., 0.1, 0.01, 0.001), the values of f(x)f(x) increase (e.g., 100, 10,000, 1,000,000).
  • The closer xx gets to 0, the larger f(x)f(x) becomes.
  • For any large value of f(x)f(x), there exists an xx very close to 0 such that f(x)f(x) equals that large value.
  • As xx approaches 0, f(x)f(x) grows without bound.
  • Strictly speaking, lim⁡x→01x2\lim_{x \to 0} \frac{1}{x^2} does not exist because f(x)f(x) becomes infinitely large.
  • Shorthand notation: lim⁡x→01x2=∞\lim_{x \to 0} \frac{1}{x^2} = \infty. This means that as xx approaches 0, the function 1x2\frac{1}{x^2} grows larger and does not tend to any specific number.
  • More generally, the value of a limit as xx approaches aa is referred to as infinity if an arbitrarily large positive value is approached or negative infinity if an arbitrarily large negative value is approached.
Example: f(x)=1x−af(x) = \frac{1}{x - a}
  • Consider the behavior of f(x)f(x) as xx approaches aa, where aa is a real number.
  • The graph of f(x)f(x) has a vertical asymptote at x=ax = a.
  • The function resembles the graph of 1x\frac{1}{x} translated horizontally by aa units.
  • lim⁡x→a−f(x)=−∞\lim_{x \to a^-} f(x) = -\infty. As xx approaches aa from the left, f(x)f(x) decreases unboundedly, tending to negative infinity.
  • lim⁡x→a+f(x)=∞\lim_{x \to a^+} f(x) = \infty. As xx approaches aa from the right, f(x)f(x) increases unboundedly, tending to infinity.
  • These limits do not exist in the traditional sense, meaning they are not real numbers.
Example: g(x)=x+1x−1g(x) = \frac{x + 1}{x - 1}
  • Analyze the behavior of the graph of g(x)g(x) as xx approaches a value aa, considering both left-hand and right-hand limits.
  • Rewrite g(x)g(x) as g(x)=1+2x−1g(x) = 1 + \frac{2}{x - 1} to understand its graph more easily.
  • The graph of g(x)g(x) has a vertical asymptote at x=1x = 1.
  • lim⁡x→1−g(x)=−∞\lim_{x \to 1^-} g(x) = -\infty. As xx approaches 1 from the left, g(x)g(x) decreases unboundedly.
  • lim⁡x→1+g(x)=∞\lim_{x \to 1^+} g(x) = \infty. As xx approaches 1 from the right, g(x)g(x) increases unboundedly.
  • These limits do not exist.

Limits at Infinity

  • In engineering, it is important to consider limits as xx approaches infinity, providing information about the limiting behavior of a function at very large values.
    *Example:
    f(x) = \frac{x^2 - 1}{x^2 + 1}
  • Consider values of xx that grow unboundedly (e.g., 10, 100, 1000, …).
  • As xx increases, the values of f(x)f(x) approach 1.
  • The function is even, meaning f(x)=f(−x)f(x) = f(-x).
  • As xx becomes very large (positive or negative), f(x)f(x) approaches 1.
  • lim⁡x→∞x2−1x2+1=1\lim_{x \to \infty} \frac{x^2 - 1}{x^2 + 1} = 1.
  • lim⁡x→−∞x2−1x2+1=1\lim_{x \to -\infty} \frac{x^2 - 1}{x^2 + 1} = 1.
  • The graph of the function has a horizontal asymptote at y=1y = 1.
Visualizing Infinity
  • In the Cartesian plane R2\mathbb{R}^2, the positive x-direction represents positive infinity, the negative x-direction represents negative infinity, and similarly for the y-direction.
Caution: Arithmetic with Infinity
  • It is incorrect to directly substitute infinity into an expression and perform arithmetic operations.
  • For example, ∞∞\frac{\infty}{\infty} is not equal to 1.
  • Arithmetic with infinity is undefined in this unit.
General Definition of a Limit at Infinity
  • Let ff be a function defined on the interval (a,∞)(a, \infty).
  • lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L if f(x)f(x) can be made arbitrarily close to LL by taking xx sufficiently large.
  • For example, lim⁡x→∞1x=0\lim_{x \to \infty} \frac{1}{x} = 0. As xx becomes larger, 1x\frac{1}{x} approaches 0.
  • Similarly, lim⁡x→−∞1x=0\lim_{x \to -\infty} \frac{1}{x} = 0.
  • Using limit laws, lim⁡x→∞1xn=0\lim_{x \to \infty} \frac{1}{x^n} = 0 for any positive integer nn.
Calculating Limits at Infinity Algebraically
  • The limit lim⁡<em>x→∞x2−1x2+1=1\lim<em>{x \to \infty} \frac{x^2 - 1}{x^2 + 1} = 1 can be calculated algebraically using the limit lim⁡</em>x→∞1x=0\lim</em>{x \to \infty} \frac{1}{x} = 0.
  • Factor out x2x^2 from the numerator and denominator:
    lim⁡<em>x→∞x2−1x2+1=lim⁡</em>x→∞x2(1−1x2)x2(1+1x2)\lim<em>{x \to \infty} \frac{x^2 - 1}{x^2 + 1} = \lim</em>{x \to \infty} \frac{x^2(1 - \frac{1}{x^2})}{x^2(1 + \frac{1}{x^2})}
  • Cancel the x2x^2 terms:
    lim⁡x→∞1−1x21+1x2\lim_{x \to \infty} \frac{1 - \frac{1}{x^2}}{1 + \frac{1}{x^2}}
  • Apply limit properties:
    lim⁡<em>x→∞1−1x21+1x2=lim⁡</em>x→∞1−lim⁡<em>x→∞1x2lim⁡</em>x→∞1+lim⁡x→∞1x2=1−01+0=1\lim<em>{x \to \infty} \frac{1 - \frac{1}{x^2}}{1 + \frac{1}{x^2}} = \frac{\lim</em>{x \to \infty} 1 - \lim<em>{x \to \infty} \frac{1}{x^2}}{\lim</em>{x \to \infty} 1 + \lim_{x \to \infty} \frac{1}{x^2}} = \frac{1 - 0}{1 + 0} = 1
More Examples

*Example:
\lim_{x \to \infty} \frac{1 - 2x^2}{x^2 + 1}

  • Multiply and divide by the highest power of xx in the expression (in this case, x2x^2).
  • lim⁡<em>x→∞1x2−21+1x2=0−21+0=−2\lim<em>{x \to \infty} \frac{\frac{1}{x^2} - 2}{1 + \frac{1}{x^2}} = \frac{0 - 2}{1 + 0} = -2 *Example: \lim{x \to \infty} \frac{\sqrt{1 + x^2}}{x}
  • Multiply and divide by xx.
  • lim⁡<em>x→∞1+x2x=lim⁡</em>x→∞1+x2x2=lim⁡x→∞1x2+1=0+1=1\lim<em>{x \to \infty} \frac{\sqrt{1 + x^2}}{x} = \lim</em>{x \to \infty} \sqrt{\frac{1 + x^2}{x^2}} = \lim_{x \to \infty} \sqrt{\frac{1}{x^2} + 1} = \sqrt{0 + 1} = 1
  • The square root function is continuous, allowing the limit to be taken inside the square root.
Homework
  • Calculate lim⁡x→−∞1−2x2x2+1\lim_{x \to -\infty} \frac{1 - 2x^2}{x^2 + 1}.
Limits of Rational Functions
  • A rational function is a fraction of two polynomials.
Examples:
Example 1:

*lim⁡x→∞x2+13x2+x\lim_{x \to \infty} \frac{x^2 + 1}{3x^2 + x}

  • The degree of the numerator and denominator are both 2.
  • Factor out x2x^2:
    lim⁡<em>x→∞x2(1+1x2)x2(3+1x)=lim⁡</em>x→∞1+lim⁡<em>x→∞1x2lim⁡</em>x→∞3+lim⁡x→∞1x=1+03+0=13\lim<em>{x \to \infty} \frac{x^2(1 + \frac{1}{x^2})}{x^2(3 + \frac{1}{x})} = \frac{\lim</em>{x \to \infty} 1 + \lim<em>{x \to \infty} \frac{1}{x^2}}{\lim</em>{x \to \infty} 3 + \lim_{x \to \infty} \frac{1}{x}} = \frac{1 + 0}{3 + 0} = \frac{1}{3}
  • When the degrees of the numerator and denominator are equal, the limit as xx approaches infinity is the ratio of the coefficients of the highest power of xx.
Example 2:

*lim⁡<em>x→∞a</em>nxn+a<em>n−1xn−1+…+a</em>1x+a<em>0b</em>nxn+b<em>n−1xn−1+…+b</em>1x+b<em>0=a</em>nbn\lim<em>{x \to \infty} \frac{a</em>n x^n + a<em>{n-1} x^{n-1} + … + a</em>1 x + a<em>0}{b</em>n x^n + b<em>{n-1} x^{n-1} + … + b</em>1 x + b<em>0} = \frac{a</em>n}{b_n}

Example 3:

*lim⁡x→∞x2+x+1x3+x\lim_{x \to \infty} \frac{x^2 + x + 1}{x^3 + x}

  • The degree of the denominator (3) is greater than the degree of the numerator (2).
    lim⁡<em>x→∞x2+x+1x3+x=lim⁡</em>x→∞1x+1x2+1x31+1x2=0+0+01+0=0\lim<em>{x \to \infty} \frac{x^2 + x + 1}{x^3 + x} = \lim</em>{x \to \infty} \frac{\frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}}{1 + \frac{1}{x^2}} = \frac{0 + 0 + 0}{1 + 0} = 0
  • When the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, the limit is always 0.

*\lim{x \to \infty} \frac{an x^n + … + a0}{bm x^m + … + b_0} = 0 \text{ if } m > n

Example 4:

*\lim{x \to \infty} \frac{an x^n + … + a0}{bm x^m + … + b_0} = \infty \text{ if } n > m

Limits Involving Exponential Functions
  • The exponential function increases faster than any integer power of xx.
  • lim⁡x→∞xnex=0\lim_{x \to \infty} \frac{x^n}{e^x} = 0 for any positive integer nn.

*Graphically, the exponential exe^x is always above any polynomial function.

More Limits Involving Exponential Functions

*lim⁡x→−∞∣x∣ne−x=0\lim_{x \to -\infty} \frac{|x|^n}{e^{-x}} = 0

*Example:
lim⁡x→−∞∣x∣e−x\lim_{x \to -\infty} \frac{|x|}{e^{-x}}

  • As xx tends to negative infinity, −x-x tends to infinity.
  • If x=−1000x = -1000, then −x=1000-x = 1000.
  • As xx approaches negative infinity, −x-x approaches infinity, and ∣x∣e−x\frac{|x|}{e^{-x}} approaches 0.
Shorthand Notation for Infinite Limits
  • Although the limit does not technically exist, the notation lim⁡x→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty is used to indicate that the function grows unboundedly.
Examples:
Case 1:

*lim⁡x→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty

  • Example: y=xy = \sqrt{x}
Case 2:

*lim⁡x→∞f(x)=−∞\lim_{x \to \infty} f(x) = -\infty

  • Example: y=−xy = -\sqrt{x}
Case 3:

*lim⁡x→−∞f(x)=∞\lim_{x \to -\infty} f(x) = \infty

  • Example: y=e−xy = e^{-x}
Case 4:

*lim⁡x→−∞f(x)=−∞\lim_{x \to -\infty} f(x) = -\infty

  • Example: y=−e−xy = -e^{-x}
  • In Angsten 90, use the notation "does not exist" when the limit is infinite, but describe the behavior of the function (e.g., grows unboundedly).
    Arithmetic with infinity (e.g., ∞−∞\infty - \infty, ∞∞\frac{\infty}{\infty}) is not well-defined.
  • However, 1∞=0\frac{1}{\infty} = 0 is an intuitive concept.

Asymptotes

Vertical Asymptotes
  • The line x=ax = a is a vertical asymptote for the curve y=f(x)y = f(x) if at least one of the following is true:
    *lim⁡<em>x→af(x)=∞\lim<em>{x \to a} f(x) = \infty *lim⁡</em>x→af(x)=−∞\lim</em>{x \to a} f(x) = -\infty
    *lim⁡<em>x→a−f(x)=∞\lim<em>{x \to a^-} f(x) = \infty *lim⁡</em>x→a−f(x)=−∞\lim</em>{x \to a^-} f(x) = -\infty
    *lim⁡<em>x→a+f(x)=∞\lim<em>{x \to a^+} f(x) = \infty *lim⁡</em>x→a+f(x)=−∞\lim</em>{x \to a^+} f(x) = -\infty

*Example:
f(x) = \frac{1}{x}

  • lim⁡x→0−f(x)=−∞\lim_{x \to 0^-} f(x) = -\infty
  • lim⁡x→0+f(x)=∞\lim_{x \to 0^+} f(x) = \infty
  • The function has a vertical asymptote at x=0x = 0.
Horizontal Asymptotes
  • The line y=Ly = L is a horizontal asymptote for the curve y=f(x)y = f(x) if:
    *lim⁡<em>x→∞f(x)=L\lim<em>{x \to \infty} f(x) = L *lim⁡</em>x→−∞f(x)=L\lim</em>{x \to -\infty} f(x) = L

*Example:
f(x) = \frac{x^2 - 1}{x^2 + 1}

  • lim⁡<em>x→∞f(x)=1\lim<em>{x \to \infty} f(x) = 1lim⁡</em>x→−∞f(x)=1\lim</em>{x \to -\infty} f(x) = 1
  • So, we can see that y=1y = 1 is a horizontal asymptote.
  • The graph of the function approaches 1 as xx approaches infinity or negative infinity.
  • A function can cross a horizontal asymptote at some points. The horizontal asymptote describes the behavior of the function at very large values of xx.