Calculus Limit Guide & Step-by-Step Rules

1. Direct Substitution

The first rule of limit evaluation is direct substitution. If $f(x)$ is continuous at $x = c$, then:

2. Resolving Indeterminate Forms

When direct substitution yields an indeterminate form, try algebraic techniques:

• Factoring: Factor numerator and denominator to cancel common zero-factors.
• Conjugate Rationalization: For radical expressions, multiply numerator and denominator by the conjugate.
• L'Hôpital's Rule: If substitution yields 0/0 or inf/inf, apply derivatives to numerator and denominator.

3. Standard Trigonometric Limits

Key trigonometric limit formulas used in calculus:

4. Limits at Infinity

For limits as $x \to \infty$, divide every term in a rational function by the highest power of $x$ in the denominator. For polynomial ratios P(x)/Q(x):

• If degree of P is less than Q, the limit is 0.
• If degree of P equals Q, the limit is the ratio of leading coefficients.
• If degree of P is greater than Q, the limit is infinite.