LHopitals Rule Calculating Limits Using Derivatives Limit Review
- Slides: 15
L’Hopital’s Rule Calculating Limits Using Derivatives
Limit Review: Evaluate each limit: We can just plug in the 1 We can’t just plug in the 1, but the expression can simplify… Then we can just plug in the 1
Limit Review: Evaluate each limit at infinity: Denominator degree higher, limit is 0 Numerator degree is higher, limit is infinity. The numerator and denominator have the same degree, the limit is the ratio of the two coefficients.
Find the limit: • We can’t use any of our previous tools. • We can’t plug in the 1, because it creates 0/0, an indeterminate form. • The expression doesn’t simplify. • It isn’t a limit at infinity with polynomial numerator and denominator.
L’Hopital’s Rule To be used when you have an Indeterminate form of a limit: Top and bottom are both 0 or top and bottom are both infinity. indeterminate forms
Indeterminate form: Top and bottom are both 0 This limit is in the form 0/0 and cannot be simplified. To evaluate this limit, we use L’Hopital’s Rule:
L’Hopital’s Rule Suppose that f and g are differentiable on the interval (a, b), except possibly at some fixedpoint c (a, b) and that g’(x) ≠ 0 on (a, b) except possibly at c. Suppose further that has the indeterminate form 0/0 or / and that Be careful! This is NOT a quotient rule. Take the derivative of the top then separately take the derivative of the bottom.
L’Hopital’s Rule Derivative of numerator Derivative of denominator
L’Hopital’s Rule The graph confirms that near 1, the graph approaches a height of 1. The graph of the function Close-up near x = 1
Example 1: Evaluate the limit. L’Hopital’s rule applies since replacing with infinity gives: Derivative of numerator Derivative of denominator
Example 1: Evaluate the limit. The graph confirms that as x continues to get larger, the graph approaches a height of 0.
Example 2: Evaluate the limit. L’Hopital’s rule applies since replacing with infinity gives: Derivative of numerator Derivative of denominator
Example 2: Evaluate the limit. The graph confirms that as x continues to get larger, the graph approaches a height of 0.
Example 3: Evaluate the limit. L’Hopital’s rule applies since replacing with infinity gives: Use L’Hopital’s rule again, since still indeterminant.
Example 3: Evaluate the limit. The graph confirms that as x continues to get larger, the graph gets higher and higher.