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. Here is the plot of

5. Here is the plot of their tangent lines

6. Zoomed in…

7. Indeterminate Forms 0/0 L’Hopital’s Rule
If f and g are differentiable, f(a)=g(a)=0, then

8. 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?

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

10. Calculate the following limits

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