1 min read•february 15, 2024
Welcome to the last key topic of unit 4! 🥳
You may recall from Unit 1 that sometimes the limits of functions evaluate to or which are indeterminate forms. Now, instead of looking for another way to manipulate the equation to try and find an answer that is not in indeterminate form, we can use our knowledge of derivatives to help us! Specifically, we can use L’Hopital’s Rule. ⭐
L’Hopital’s Rule states that if or ,
Basically, the rule states that we can evaluate the limits of indeterminate forms using their derivatives!
Let’s try a practice problem together! Evaluate the following limit.
Plugging into results in the indeterminate form . This signals to us that we should use L’Hopital’s Rule.
When completing a free-response question on the AP exam, it is important to show the limits of f(x), the numerator, and g(x), the denominator, are separately equal to the needed parameters. Let’s go ahead and do that now!
Since and , L’Hopital’s Rule can be applied. Be sure to write this statement out before actually applying this rule.
Now, we can take the derivatives and get into L’Hopital’s Rule.
In conclusion, we know that this limit…
Great work! Doesn’t L’Hopital’s Rule save so much time? 🙃
Here are some problems you can practice applying L’Hopital’s Rule on!
Evaluate the following limits. Imagine these are free-response questions and you have to check conditions!
Plugging into results in the indeterminate form . Since the expression involves mixed function types, it is not possible to manipulate it algebraically in any way to find the limits. Therefore, we should use L’Hopital’s Rule.
But first, show that the limits are separately equal to 0.
Since and , L’Hopital’s Rule can be applied.
In conclusion…
Plugging into results in the indeterminate form . Therefore, we should use L’Hopital’s Rule.
Since and , L’Hopital’s Rule can be applied.
Therefore…
Remember, L'Hopital's Rule provides a powerful method for evaluating limits involving indeterminate forms. By taking derivatives of the numerator and denominator, it helps simplify complex limits, leading to easier evaluation.
Happy calculus studying! 📚
Indeterminate Forms
: Indeterminate forms are mathematical expressions that cannot be evaluated directly because they result in an ambiguous or undefined value. These forms often arise when evaluating limits of functions.L'Hopital's Rule
: L'Hopital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of the ratio of two functions is an indeterminate form, then taking the derivative of both functions and evaluating the limit again can help determine the original limit.1 min read•february 15, 2024
Welcome to the last key topic of unit 4! 🥳
You may recall from Unit 1 that sometimes the limits of functions evaluate to or which are indeterminate forms. Now, instead of looking for another way to manipulate the equation to try and find an answer that is not in indeterminate form, we can use our knowledge of derivatives to help us! Specifically, we can use L’Hopital’s Rule. ⭐
L’Hopital’s Rule states that if or ,
Basically, the rule states that we can evaluate the limits of indeterminate forms using their derivatives!
Let’s try a practice problem together! Evaluate the following limit.
Plugging into results in the indeterminate form . This signals to us that we should use L’Hopital’s Rule.
When completing a free-response question on the AP exam, it is important to show the limits of f(x), the numerator, and g(x), the denominator, are separately equal to the needed parameters. Let’s go ahead and do that now!
Since and , L’Hopital’s Rule can be applied. Be sure to write this statement out before actually applying this rule.
Now, we can take the derivatives and get into L’Hopital’s Rule.
In conclusion, we know that this limit…
Great work! Doesn’t L’Hopital’s Rule save so much time? 🙃
Here are some problems you can practice applying L’Hopital’s Rule on!
Evaluate the following limits. Imagine these are free-response questions and you have to check conditions!
Plugging into results in the indeterminate form . Since the expression involves mixed function types, it is not possible to manipulate it algebraically in any way to find the limits. Therefore, we should use L’Hopital’s Rule.
But first, show that the limits are separately equal to 0.
Since and , L’Hopital’s Rule can be applied.
In conclusion…
Plugging into results in the indeterminate form . Therefore, we should use L’Hopital’s Rule.
Since and , L’Hopital’s Rule can be applied.
Therefore…
Remember, L'Hopital's Rule provides a powerful method for evaluating limits involving indeterminate forms. By taking derivatives of the numerator and denominator, it helps simplify complex limits, leading to easier evaluation.
Happy calculus studying! 📚
Indeterminate Forms
: Indeterminate forms are mathematical expressions that cannot be evaluated directly because they result in an ambiguous or undefined value. These forms often arise when evaluating limits of functions.L'Hopital's Rule
: L'Hopital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of the ratio of two functions is an indeterminate form, then taking the derivative of both functions and evaluating the limit again can help determine the original limit.© 2024 Fiveable Inc. All rights reserved.
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