Evaluating limits algebraically | Calculus

Limit laws and direct substitution

Limits behave well with arithmetic: the limit of a sum, difference, product or quotient is the sum, difference, product or quotient of the limits (as long as you don't divide by zero).

For polynomials, rational functions, roots, exponentials, logs and trig functions, at a point in the domain, you can simply substitute:

\lim_{x \to 2}\frac{x^2 + 1}{x + 3} = \frac{5}{5} = 1

Indeterminate forms

Substitution sometimes gives an expression with no fixed value, called an indeterminate form:

\frac{0}{0} \qquad \frac{\infty}{\infty} \qquad \infty - \infty \qquad 0 \cdot \infty \qquad 1^\infty \qquad 0^0 \qquad \infty^0

$\frac{0}{0}$ doesn't mean the limit is $$0$$ , $$1$$ or undefined: it means more work is needed.

Factoring

Rationalizing

Complex fractions

Limits at infinity of rational functions

Divide every term by the highest power of $$x$$ in the denominator, or just compare degrees.

Practice

Quick check

Substituting into a limit gives $\frac{0}{0}$ . What can you conclude?

Quick check

What is $\lim_{x \to -\infty} \frac{6x^2 - 1}{2x^2 + x}$ ?