Limits at Infinity and Infinite Limits – Calculus Study Notes (Section 2.6)

Limits at Infinity and Infinite Limits

Concepts

This section explores the behavior of functions as the input variable approaches infinity or negative infinity, as well as the concept of infinite limits. Understanding these ideas is crucial for analyzing end behavior and asymptotic properties of functions.

• Limits at Infinity: The notation $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ means that as $x$ increases or decreases without bound, the function $f(x)$ approaches the value $L$.

• Graphical Interpretation: To determine limits at infinity, observe the graph's behavior as $x$ moves far to the right (positive infinity) or left (negative infinity). Check if the graph flattens out to a horizontal line (horizontal asymptote).

• Theorem:

  • $\lim_{x \to \infty} \frac{1}{x} = 0$

• Limit Laws: All standard limit laws apply to limits at infinity (see Section 2.2 for details).

• Steps for Computing $\lim_{x \to \infty} \frac{f(x)}{g(x)}$ (for rational expressions):

  1. Find the highest power of $x$ in both the numerator and denominator.

  2. Factor out the largest power of $x$ from both numerator and denominator.

  3. Simplify the expression.

  4. Use the theorem above to evaluate the limit.

• Infinite Limits: An infinite limit occurs when $\lim_{x \to a} f(x) = \infty$ or $\lim_{x \to a} f(x) = -\infty$. This means the function increases or decreases without bound as $x$ approaches $a$.

• Determining Infinite Limits: To determine whether a limit is $+\infty$ or $-\infty$, try a value close to $a$ and observe the sign of the function.

Examples and Applications

• Example 1: $\lim_{x \to \infty} \frac{2x^2 + 3x + 4}{x^2 - 5x + 6}$

  • Highest power of $x$ is $x^2$ in both numerator and denominator.

  • Divide numerator and denominator by $x^2$:

  • $\lim_{x \to \infty} \frac{2 + \frac{3}{x} + \frac{4}{x^2}}{1 - \frac{5}{x} + \frac{6}{x^2}} = \frac{2}{1} = 2$

• Example 2: $\lim_{x \to 3^+} \frac{1}{x-3}$

  • As $x$ approaches 3 from the right, $x-3$ is a small positive number, so the limit is $+\infty$.

Practice Problems

For the function $f$ whose graph is given, determine the following limits:

• (a) $\lim_{x \to -\infty} f(x)$

• (b) $\lim_{x \to \infty} f(x)$

• (c) $\lim_{x \to 2^-} f(x)$

• (d) $\lim_{x \to 2^+} f(x)$

• (e) $\lim_{x \to 2} f(x)$

• (f) $f(2)$

Note: Use the graph to determine the left-hand and right-hand limits, as well as the function value at $x=2$.

Find the limits at infinity:

• (a) $\lim_{x \to \infty} \frac{2x^2 + 3x + 4}{x^2 - 5x + 6}$

• (b) $\lim_{x \to -\infty} \frac{2x^2 + 3x + 4}{x^2 - 5x + 6}$

• (c) $\lim_{x \to \infty} \frac{3x^2 + 2x + 3}{2x^2 + 5x^2 + 5x}$

• (d) $\lim_{x \to \infty} \left(1 - \frac{5}{x}\right)^7$

• (e) $\lim_{x \to \infty} \frac{\sqrt{x^2 + 7x} - x}{8x^2 - 3}$

• (f) $\lim_{x \to \infty} \frac{\sin(2x)}{x}$

Find the following infinite limits:

• (a) $\lim_{x \to 5^+} \frac{1}{x-5}$

• (b) $\lim_{x \to 5^-} \frac{1}{(x-5)^2}$

• (c) $\lim_{x \to 7^-} \frac{5}{(x-7)^3}$

• (d) $\lim_{x \to 2^+} \frac{1}{x^2 - 4}$

• (e) $\lim_{x \to 2^-} \frac{1}{x^2 - 4}$

• (f) $\lim_{x \to -2^+} \frac{1}{x^2 - 4}$

• (g) $\lim_{x \to -2^-} \frac{1}{x^2 - 4}$

Summary Table: Types of Limits at Infinity

Type	Form	Typical Result	Example
Limit at Infinity (Rational)	$\lim_{x \to \infty} \frac{P(x)}{Q(x)}$	Depends on degrees of $P$ and $Q$	$\lim_{x \to \infty} \frac{2x^2}{x^2+1} = 2$
Infinite Limit	$\lim_{x \to a} f(x) = \infty$ or $-\infty$	Function increases/decreases without bound	$\lim_{x \to 0^+} \frac{1}{x} = \infty$
Oscillating Limit	$\lim_{x \to \infty} \sin x$	Does not exist	$\lim_{x \to \infty} \sin x$

Additional info: For rational functions, if the degree of the numerator is less than the denominator, the limit at infinity is 0. If the degrees are equal, the limit is the ratio of leading coefficients. If the numerator's degree is higher, the limit is infinite (or does not exist).