L’Hospital’s Rule, Growth, and Dominance

1. L’Hospital’s Rule, Growth, and Dominance

2. Consider the following limit • What do we get when we evaluate it? • We can use local linearity • Let’s look at the graphs

3. Here is the plot of

4. Indeterminate Forms 0/0 L’Hopital’s Rule If f and g are differentiable, f(a)=g(a)=0, then This can be shown by using the functions’ tangent line approximations

5. Indeterminate Forms∞/∞ • In this case we may want to know which goes to infinity faster, the numerator or the denominator? Or do they go at about the same rate? • This can be shown using the indeterminate form 0/0

6. L’Hopital’s Rule • If f and g are differentiable, - When limf(a)=g( , and • or • When g’(a)≠0, t hen • It can be shown that • Where a may be ±∞ • Provided the limit on the right exists

7. Indeterminate Forms • There are other ‘indeterminate forms’ • Each of these can be changed to be 0/0 or ∞/∞ • Examples

8. Calculate the following limits

9. We say that g dominates f as x→∞ if • Check that x½ dominates lnx as x→∞ • Check that 2x dominates x2 as x→∞