Essential Concepts
- L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form [latex]\frac{0}{0}[/latex] or [latex]\frac{\infty}{\infty}[/latex] arises.
- L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form [latex]\frac{0}{0}[/latex] or [latex]\frac{\infty}{\infty}[/latex].
- The exponential function [latex]e^x[/latex] grows faster than any power function [latex]x^p[/latex], [latex]p>0[/latex].
- The logarithmic function [latex]\ln x[/latex] grows more slowly than any power function [latex]x^p[/latex], [latex]p>0[/latex].
Glossary
- indeterminate forms
- when evaluating a limit, the forms [latex]0/0[/latex], [latex]\infty / \infty[/latex], [latex]0 \cdot \infty[/latex], [latex]\infty -\infty[/latex], [latex]0^0[/latex], [latex]\infty^0[/latex], and [latex]1^{\infty}[/latex] are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is
- L’Hôpital’s rule
- if [latex]f[/latex] and [latex]g[/latex] are differentiable functions over an interval [latex]a[/latex], except possibly at [latex]a[/latex], and [latex]\underset{x\to a}{\lim} f(x)=0=\underset{x\to a}{\lim} g(x)[/latex] or [latex]\underset{x\to a}{\lim} f(x)[/latex] and [latex]\underset{x\to a}{\lim} g(x)[/latex] are infinite, then [latex]\underset{x\to a}{\lim}\dfrac{f(x)}{g(x)}=\underset{x\to a}{\lim}\dfrac{f^{\prime}(x)}{g^{\prime}(x)}[/latex], assuming the limit on the right exists or is [latex]\infty [/latex] or [latex]−\infty [/latex]
