Summary of the Limit Laws

The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time.
For polynomials and rational functions, [latex]\underset{x\to a}{\lim}f(x)=f(a)[/latex].
You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction.
The Squeeze Theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known.

Key Equations

Basic Limit Results
[latex]\underset{x\to a}{\lim}x=a[/latex]
[latex]\underset{x\to a}{\lim}c=c[/latex]
Important Limits
[latex]\underset{\theta \to 0}{\lim} \sin \theta =0[/latex]
[latex]\underset{\theta \to 0}{\lim} \cos \theta =1[/latex]
[latex]\underset{\theta \to 0}{\lim}\frac{\sin \theta}{\theta}=1[/latex]
[latex]\underset{\theta \to 0}{\lim}\frac{1- \cos \theta}{\theta}=0[/latex]

infinite limit: A function has an infinite limit at a point [latex]a[/latex] if it either increases or decreases without bound as it approaches [latex]a[/latex]

intuitive definition of the limit: If all values of the function [latex]f(x)[/latex] approach the real number [latex]L[/latex] as the values of [latex]x(\ne a)[/latex] approach [latex]a[/latex], [latex]f(x)[/latex] approaches [latex]L[/latex]

one-sided limit: A one-sided limit of a function is a limit taken from either the left or the right

vertical asymptote: A function has a vertical asymptote at [latex]x=a[/latex] if the limit as [latex]x[/latex] approaches [latex]a[/latex] from the right or left is infinite