- Determine limits and predict how functions behave as x increases or decreases indefinitely
- Identify and distinguish horizontal and slanting lines that a graph approaches but never touches
- Use a function’s derivatives to accurately sketch its graph
Limits at Infinity
The Main Idea
- Limits at Infinity:
- Describes behavior of [latex]f(x)[/latex] as [latex]x[/latex] approaches [latex]\infty[/latex] or [latex]-\infty[/latex]
- Notation: [latex]\lim_{x \to \infty} f(x) = L[/latex] or [latex]\lim_{x \to -\infty} f(x) = L[/latex]
- Horizontal Asymptotes:
- Line [latex]y = L[/latex] where [latex]f(x)[/latex] approaches [latex]L[/latex] as [latex]x \to \pm\infty[/latex]
- Function may cross horizontal asymptote multiple times
- Infinite Limits at Infinity:
- When [latex]f(x)[/latex] grows without bound as [latex]x \to \pm\infty[/latex]
- Notation: [latex]\lim_{x \to \infty} f(x) = \infty[/latex] or [latex]\lim_{x \to \infty} f(x) = -\infty[/latex]
- Formal Definitions:
- For finite limits: [latex]\forall \varepsilon > 0, \exists N > 0: |f(x) - L| < \varepsilon[/latex] when [latex]x > N[/latex]
- For infinite limits: [latex]\forall M > 0, \exists N > 0: f(x) > M[/latex] when [latex]x > N[/latex]
- Strategies for Evaluation:
- Use algebraic limit laws
- Apply the squeeze theorem
- Analyze rational functions by comparing highest degree terms
Evaluate [latex]\underset{x\to −\infty}{\lim}\left(3+\frac{4}{x}\right)[/latex] and [latex]\underset{x\to \infty }{\lim}\left(3+\frac{4}{x}\right)[/latex]. Determine the horizontal asymptotes of [latex]f(x)=3+\frac{4}{x}[/latex], if any.
Use the formal definition of limit at infinity to prove that [latex]\underset{x\to \infty }{\lim}\left(3 - \dfrac{1}{x^2}\right)=3[/latex].
Use the formal definition of infinite limit at infinity to prove that [latex]\underset{x\to \infty }{\lim}3x^2=\infty[/latex].
End Behavior
The Main Idea
- End Behavior:
- Describes how a function behaves as [latex]x \to \pm\infty[/latex]
- Three possible outcomes: approach a finite value, approach infinity, or oscillate
- Polynomial End Behavior:
- Determined by the highest degree term
- Even degree: same behavior at both ends
- Odd degree: opposite behavior at each end
- Rational Function End Behavior:
- Depends on the degree relationship between numerator and denominator
- Three cases: horizontal asymptote, zero asymptote, or unbounded growth
- Horizontal Asymptotes:
- [latex]y = L[/latex] where [latex]\lim_{x \to \pm\infty} f(x) = L[/latex]
- For rational functions [latex]\frac{p(x)}{q(x)}[/latex]:
- If deg(p) < deg(q): [latex]y = 0[/latex]
- If deg(p) = deg(q): [latex]y = \frac{a_n}{b_m}[/latex] (ratio of leading coefficients)
- If deg(p) > deg(q): no horizontal asymptote
- Transcendental Function End Behavior:
- Trigonometric: oscillate (no limit)
- Exponential ([latex]e^x[/latex]): [latex]\to 0[/latex] as [latex]x \to -\infty[/latex], [latex]\to \infty[/latex] as [latex]x \to \infty[/latex]
- Natural log: [latex]\to -\infty[/latex] as [latex]x \to 0^+[/latex], [latex]\to \infty[/latex] as [latex]x \to \infty[/latex]
Let [latex]f(x)=-3x^4[/latex]. Find [latex]\underset{x\to \infty }{\lim}f(x)[/latex].
Evaluate [latex]\underset{x\to \pm \infty }{\lim}\dfrac{3x^2+2x-1}{5x^2-4x+7}[/latex] and use these limits to determine the end behavior of [latex]f(x)=\dfrac{3x^2+2x-1}{5x^2-4x+7}[/latex].
Evaluate [latex]\underset{x\to \infty }{\lim}\frac{\sqrt{3x^2+4}}{x+6}[/latex].
Find the limits as [latex]x\to \infty[/latex] and [latex]x\to −\infty[/latex] for [latex]f(x)=\dfrac{(3e^x-4)}{(5e^x+2)}[/latex].
Drawing Graphs of Functions
Sketch a graph of [latex]f(x)=(x-1)^3 (x+2)[/latex]
Sketch a graph of [latex]f(x)=\dfrac{3x+5}{8+4x}[/latex]
Find the oblique asymptote for [latex]f(x)=\dfrac{3x^3-2x+1}{2x^2-4}[/latex]
Consider the function [latex]f(x)=5-x^{\frac{2}{3}}[/latex]. Determine the point on the graph where a cusp is located. Determine the end behavior of [latex]f[/latex].