Limits and Continuity Cheat Sheet

Understanding limits and continuity is fundamental to calculus, enabling the analysis of function behavior at specific points and over intervals. This involves evaluating limits, identifying discontinuities, and determining where functions are continuous.

Core Principles

Limit Existence: A limit exists at a point if and only if the left-hand limit and the right-hand limit are equal.
Continuity: A function is continuous at a point 'c' if the limit as x approaches 'c' exists, the function is defined at 'c', and the limit equals the function's value at 'c'.
Average Rate of Change: Calculated as the change in the function's value divided by the change in the input variable over a given interval.
Vertical Asymptotes: Occur where the limit of a function approaches infinity or negative infinity as the input approaches a specific value.
Holes: Occur at points where a function is undefined but the limit exists, often due to a removable discontinuity.

Action Steps

To evaluate a limit of a rational function, substitute the value x approaches. If the result is indeterminate (e.g., 0/0), simplify the expression (e.g., by factoring) before substituting.
To find vertical asymptotes, set the denominator of a rational function to zero and solve for x. Verify these values by checking if the limit as x approaches these points is $\pm\infty$ or DNE.
To identify holes, find values of x that make both the numerator and denominator of a rational function zero. Simplify the function by canceling common factors and evaluate the limit at that point.
To determine continuity of a polynomial function, check if it's defined for all real numbers. For rational functions, identify points where the denominator is zero.
To find the limit of a polynomial as x approaches $\pm\infty$, consider the term with the highest degree.

Formulas

Average Rate of Change: $ \frac{g(t_2) - g(t_1)}{t_2 - t_1} $
Limit of a rational function at infinity: For $ \lim_{x\to\infty} \frac{a_n x^n + ...}{b_m x^m + ...} $, if n < m, the limit is 0; if n = m, the limit is $\frac{a_n}{b_m}$; if n > m, the limit is $\pm\infty$.

Key Terms

Limit: The value that a function or sequence 'approaches' as the input or index approaches some value.
Continuity: A function is continuous on an interval if it is continuous at every number in the interval.
Average Rate of Change: The slope of the secant line connecting two points on a function's graph.
Vertical Asymptote: A vertical line x=a where the function's output approaches infinity or negative infinity as the input approaches 'a'.
Hole: A point of removable discontinuity in a function's graph.

Real World Examples

The function $g(t) = 2 + \cos t$ over the interval $[0, \pi]$: The average rate of change is calculated as $\frac{g(\pi) - g(0)}{\pi - 0} = \frac{(2 + \cos \pi) - (2 + \cos 0)}{\pi} = \frac{(2 - 1) - (2 + 1)}{\pi} = \frac{1 - 3}{\pi} = -\frac{2}{\pi}$.
The function $f(x) = \frac{x^2 - 4}{x^3 + 5x^2 + 6x}$: Vertical asymptotes are at x=0 and x=-3, verified by limits approaching $\pm\infty$. A hole exists at x=-2, where the limit is 2.