Limits at Infinity, Horizontal Asymptotes, and Vertical Asymptotes

End Behavior and Infinite Limits

• Definition of End Behavior:

  • Analyzing limits where $x$ grows unbounded evaluates the end behavior of a function.

  • End behavior describes the ultimate long-term trajectory of a function as input values $x$ approach positive infinity ($x \rightarrow \infty$) or negative infinity ($x \rightarrow -\infty$).

• Unbounded Growth Examples:

  • Example 1:

  • Problem: Evaluate $\lim_{x \rightarrow \infty} 2x^2$

  • Analysis: As $x$ increases without bound towards $\infty$, $x^2$ grows infinitely large. Multiplying by $2$ maintains this unbounded positive growth.

  • Result: $\lim_{x \rightarrow \infty} 2x^2 = \infty$ (The limit does not exist as a finite real number).

  • Example 2:

  • Problem: Evaluate $\lim_{x \rightarrow \infty} \frac{1}{x}$

  • Analysis: As $x$ approaches $\infty$, the fraction $\frac{1}{x}$ has a fixed numerator and an infinitely growing denominator, driving the quotient arbitrarily close to $0$.

  • Result: $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$

  • Concept: The finite real number $0$ that the function approaches is the value of the limit, which defines a horizontal line that the graph approaches at extreme values of $x$.

Limits at Infinity and Horizontal Asymptotes

• Formal Theorems for Limits at Infinity:

  • For any positive real number $n$:     $\lim_{x \rightarrow \infty} \frac{1}{x^n} = 0$     $\lim_{x \rightarrow -\infty} \frac{1}{x^n} = 0$

• Definition of Horizontal Asymptote:

  • The line $Y = L$ (or $y = L$) is defined as a horizontal asymptote of the function $y = f(x)$ if either of the following statement conditions holds true:     $\lim_{x \rightarrow \infty} f(x) = L$     $\lim_{x \rightarrow -\infty} f(x) = L$

• Worked Examples for Rational and Exponential Functions:

  • Example 3:

  • Problem: Find $\lim_{x \rightarrow \infty} \frac{8x + 6}{3x - 1}$

  • Procedure: Divide every term in the numerator and denominator by the highest power of $x$ present in the denominator (which is $x^1$):       $\lim_{x \rightarrow \infty} \frac{\frac{8x}{x} + \frac{6}{x}}{\frac{3x}{x} - \frac{1}{x}} = \lim_{x \rightarrow \infty} \frac{8 + \frac{6}{x}}{3 - \frac{1}{x}}$

  • Evaluation: Applying the rule $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$ gives:       $\frac{8 + 0}{3 - 0} = \frac{8}{3}$

  • Conclusion: The limit is $\frac{8}{3}$, and the function has a horizontal asymptote at $y = \frac{8}{3}$.

  • Example 4:

  • Problem: Find $\lim_{x \rightarrow \infty} \frac{8x + 6}{3x^2 - 1}$

  • Procedure: Divide all terms by the highest power of $x$ in the denominator ($x^2$):       $\lim_{x \rightarrow \infty} \frac{\frac{8x}{x^2} + \frac{6}{x^2}}{\frac{3x^2}{x^2} - \frac{1}{x^2}} = \lim_{x \rightarrow \infty} \frac{\frac{8}{x} + \frac{6}{x^2}}{3 - \frac{1}{x^2}}$

  • Evaluation: Substituting limits for each reciprocal power:       $\frac{0 + 0}{3 - 0} = \frac{0}{3} = 0$

  • Conclusion: The limit is $0$, corresponding to a horizontal asymptote at $y = 0$.

  • Example 5 (Biological Application - Female Arctic Foxes):

  • Context: The age-weight relationship of female Arctic Foxes caught in Svalbard, Norway, can be estimated by the function:       $M(t) = 3102 e^{-e^{-0.022(t - 56)}}$       where $t$ represents the age of the foxes in days, and $M(t)$ represents the weight of the foxes in grams.

  • Objective: Use $M(t)$ to estimate the largest size (weight) that a female fox can attain as age $t$ grows indefinitely ($t \rightarrow \infty$).

  • Procedure:

    • Evaluate the inner exponent as $t \rightarrow \infty$:         $-0.022(t - 56) \rightarrow -\infty$

    • Evaluate the secondary exponential term:         $e^{-0.022(t - 56)} \rightarrow 0$

    • Substitute back into the outer exponential structure:         $-e^{-0.022(t - 56)} \rightarrow -0 = 0$

    • Compute the outer exponential function value:         $e^0 = 1$

    • Compute the final limit of weight $M(t)$:         $\lim_{t \rightarrow \infty} M(t) = 3102 \times 1 = 3102\text{ grams}$

  • Conclusion: The maximum achievable weight (largest size) for a female Arctic Fox under this model is $3102\text{ grams}$.

Shortcut Rules for Calculating Limits of Rational Functions as x Approaches Infinity

• Rational Function Structure:

  • For a rational function defined as $f(x) = \frac{N(x)}{D(x)}$, where $N(x)$ is the numerator polynomial and $D(x)$ is the denominator polynomial:

• Rule 1: Higher Degree in Denominator:

  • When the degree of the denominator $D(x)$ is strictly larger than the degree of the numerator $N(x)$:     $\lim_{x \rightarrow \infty} \frac{N(x)}{D(x)} = 0$

• Rule 2: Higher Degree in Numerator:

  • When the degree of the numerator $N(x)$ is strictly larger than the degree of the denominator $D(x)$:     $\lim_{x \rightarrow \infty} \frac{N(x)}{D(x)} = \infty \text{ or } -\infty$

• Rule 3: Equal Degrees:

  • When the degree of the denominator $D(x)$ is equal to the degree of the numerator $N(x)$:     $\lim_{x \rightarrow \infty} \frac{N(x)}{D(x)} = \text{the quotient of the leading coefficients}$

Vertical Asymptotes and Infinite Limits at Finite Points

• Evaluation of Infinite Limits at a Point:

  • Example 6:

  • Problem: Find $\lim_{x \rightarrow 0} \frac{1}{20x^2}$

  • Procedure and Analysis:

    • Evaluate the behavior of the expression as $x$ approaches $0$ from both sides ($x \rightarrow 0^+$ and $x \rightarrow 0^-$).

    • Since $x^2 > 0$ for all $x \neq 0$, the denominator term $20x^2$ remains strictly positive and approaches $0$

    • Taking the reciprocal of positive values approaching zero yields arbitrarily large positive values:         $\lim_{x \rightarrow 0^+} \frac{1}{20x^2} = \infty$         $\lim_{x \rightarrow 0^-} \frac{1}{20x^2} = \infty$

  • Result: $\lim_{x \rightarrow 0} \frac{1}{20x^2} = \infty$ (Does Not Exist, DNE).

  • Graphical Behavior at $x = 0$: At $x = 0$, the graph shoots vertically upwards towards positive infinity ($\infty$) from both the left and right sides.

• Formal Definition of Vertical Asymptote:

  • The vertical line $x = a$ is called a vertical asymptote of the function $y = f(x)$ if at least one of the following conditions is true:

  • $\lim_{x \rightarrow a} f(x) = \infty, \text{DNE}$

  • $\lim_{x \rightarrow a} f(x) = -\infty, \text{DNE}$

  • $\lim_{x \rightarrow a^-} f(x) = \infty, \text{DNE}$

  • $\lim_{x \rightarrow a^-} f(x) = -\infty, \text{DNE}$

  • $\lim_{x \rightarrow a^+} f(x) = \infty, \text{DNE}$

  • $\lim_{x \rightarrow a^+} f(x) = -\infty, \text{DNE}$

• Comprehensive Asymptote Identification:

  • Example 7:

  • Problem: Identify all asymptotes of the function $f(x) = \frac{2x - 3}{x + 1}$

  • Vertical Asymptote Analysis:

    • Set the denominator equal to zero to find candidate vertical discontinuities:         $x + 1 = 0 \implies x = -1$

    • Test one-sided limits as $x \rightarrow -1$:

    • As $x \rightarrow -1^+$, numerator approaches $2(-1) - 3 = -5$, and denominator approaches $0^+$ (small positive numbers), giving:           $\lim_{x \rightarrow -1^+} \frac{2x - 3}{x + 1} = -\infty$

    • As $x \rightarrow -1^-$, numerator approaches $-5$, and denominator approaches $0^-$ (small negative numbers), giving:           $\lim_{x \rightarrow -1^-} \frac{2x - 3}{x + 1} = \infty$

    • Conclusion for Vertical Asymptote: The vertical line $x = -1$ is a vertical asymptote.

  • Horizontal Asymptote Analysis:

    • Calculate limits at infinity:         $\lim_{x \rightarrow \infty} \frac{2x - 3}{x + 1} = \lim_{x \rightarrow \infty} \frac{2 - \frac{3}{x}}{1 + \frac{1}{x}} = \frac{2 - 0}{1 + 0} = 2$         $\lim_{x \rightarrow -\infty} \frac{2x - 3}{x + 1} = \lim_{x \rightarrow -\infty} \frac{2 - \frac{3}{x}}{1 + \frac{1}{x}} = \frac{2 - 0}{1 + 0} = 2$

    • Conclusion for Horizontal Asymptote: The line $y = 2$ is a horizontal asymptote.