Indeterminate Forms — Formula Sheet

An indeterminate expression is one whose value cannot be precisely determined merely from the limiting behaviour of its individual parts.

0/0, ∞/∞, 0·∞, ∞−∞, 0^0, ∞^0, 1^∞

L'Hôpital's Rule for 0/0 Form: If lim_{x→a} f(x)=0 and lim_{x→a} g(x)=0, and lim_{x→a}[f'(x)/g'(x)] = l, then lim_{x→a}[f(x)/g(x)] = lim_{x→a}[f'(x)/g

lim_{x→a}[f(x)/g(x)] with f(x)→0 and g(x)→0

L'Hôpital's Rule for ∞/∞ Form: If lim_{x→a} f(x)=∞ and lim_{x→a} g(x)=∞, and lim_{x→a}[f'(x)/g'(x)] = l, then lim_{x→a}[f(x)/g(x)] = lim_{x→a}[f'(x)/g

0·∞ is an indeterminate form

Step I: Write the form as 0·∞

Step II: Convert to 0/0 or ∞/∞

Step III: Apply L'Hôpital's Rule

The standard method is to set y=f(x)^{g(x)}, take logarithms, solve the limit of log y, and then exponentiate.

Because log is continuous in its domain, if lim(log y)=l then lim(y)=e^l.