Answer:
General Formulas and Concepts:
<u>Algebra II</u>
- Natural logarithms ln and Euler's number e
- Logarithmic Property [Exponential]:
<u>Calculus</u>
Limits
- Right-Side Limit:
- Left-Side Limit:
Limit Rule [Variable Direct Substitution]:
L’Hopital’s Rule:
Differentiation
- Derivatives
- Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Step-by-step explanation:
We are given the following limit:
Substituting in <em>x</em> = 0 using the limit rule, we have an indeterminate form:
We need to rewrite this indeterminate form to another form to use L'Hopital's Rule. Let's set our limit as a function:
Take the ln of both sides:
Rewrite the limit by including the ln in the inside:
Rewrite the limit once more using logarithmic properties:
Rewrite the limit again:
Substitute in <em>x</em> = 0 again using the limit rule, we have an indeterminate form in which we can use L'Hopital's Rule:
Apply L'Hopital's Rule:
Simplify:
Redefine the limit:
Substitute in <em>x</em> = 0 once more using the limit rule:
Evaluating it, we have:
Substitute in the limit value:
e both sides:
Simplify:
And we have our final answer.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits