Continuity, infinity, and l'Hospital
Three tools finish the limit toolkit: when a function is unbroken, where it settles at infinity, and the shortcut for indeterminate forms.
Continuity
A function is continuous at a point a if the limit as x approaches a exists, the function value f(a) exists, and the two are equal. In words: no hole, no jump, and the limit actually lands on the point. Polynomials are continuous everywhere, and quotients of continuous functions are continuous wherever the denominator is nonzero.
\lim_{x \to a} f(x) = f(a)Limits at infinity
Writing x approaching infinity asks what value the outputs approach for large x. For rational functions, divide the numerator and denominator by the highest power of x in the denominator, then let small terms vanish. The result gives any horizontal asymptote of the graph.
\lim_{x \to \infty} \frac{3x^2 + 5x}{x^2 - 2} = 3l'Hospital's rule
When a limit produces 0/0 or infinity over infinity, the limit of the ratio of derivatives equals the original limit, provided the new limit exists. For example, the limit of sin x over x as x approaches 0 becomes the limit of cos x over 1, which equals 1. Always check the indeterminate form first; the rule is invalid otherwise.
\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1Flashcard
Three conditions for continuity at a
Check your understanding
What is the limit of (5x + 2)/(2x + 7) as x approaches infinity?