Find the limit of the formula given

General Formulas and Concepts:

Algebra II

Natural logarithms ln and Euler's number eLogarithmic Property [Exponential]:                                                            

Calculus

Limits

Right-Side Limit:                                                                                             Left-Side Limit:                                                                                              

Limit Rule [Variable Direct Substitution]:                                                            

L’Hopital’s Rule:                                                                                                    

Differentiation

DerivativesDerivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹  

Step-by-step explanation:

We are given the following limit:

Substituting in x = 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 x = 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 x = 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

Jjutsclopyrsvnkiu it’s c

Lol idk ask google. ejie