L'Hopitals Rule with derivatives?

In summary, the conversation discusses evaluating the limit of a function and using derivatives to simplify the expression. The use of the chain rule and the fact that the derivative of e^x is e^x is suggested to solve the problem. The conversation ends with a suggestion to try rewriting 4^x as e^f(x) and using logarithms to solve it.
  • #1
m0286
63
0
Question is:
Evaluate the following limits
lim x--> 0 4^x - e^2x / 2x
So i take derivatives
but that's where I am confused... Whats the derivate of 4^x... x4^-x?
and is the derivative of -e^2x -2e^x?
So then that leaves me with 4x^-x -2e^x /2which is 0... so do i do derivatives again? It seems like the x's won't go away?
 
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  • #2
You can use the chain rule and the fact that the derivative of e^x is e^x to answer both questions. That is, the derivative of e^f(x) is e^f(x)*f'(x). Try to rewrite 4^x as e^f(x). (use logs)
 
  • #3
[tex]f(x)=4^x[/tex]
[tex]\ln(f(x))=x\ln(4)[/tex]
[tex]\frac{1}{f(x)}*f'(x)=\ln(4)[/tex]

You can take it from there.
 

1. What is L'Hopital's Rule and when is it used?

L'Hopital's Rule is a mathematical theorem that is used to evaluate limits involving indeterminate forms, such as 0/0 or ∞/∞. It is typically used when the direct evaluation of the limit is not possible or results in an indeterminate form.

2. How does L'Hopital's Rule work?

L'Hopital's Rule states that for a limit of f(x)/g(x) as x approaches a, if the limit of the quotient of the derivatives of f(x) and g(x) also exists as x approaches a, then this derivative limit is equal to the original limit. In other words, we can take the derivative of both the numerator and denominator and evaluate the limit again.

3. What are the conditions for using L'Hopital's Rule?

The conditions for using L'Hopital's Rule are: 1) the limit must be an indeterminate form of 0/0 or ∞/∞, 2) the functions f(x) and g(x) must be differentiable in a neighborhood of the point a, and 3) the limit of the quotient of the derivatives of f(x) and g(x) must exist as x approaches a.

4. Can L'Hopital's Rule be used for limits at infinity?

Yes, L'Hopital's Rule can be used for limits at infinity. In this case, we take the limit as x approaches infinity (or negative infinity) instead of a specific point a. The conditions for using L'Hopital's Rule still apply.

5. Are there any alternatives to using L'Hopital's Rule for evaluating indeterminate forms?

Yes, there are other methods for evaluating indeterminate forms, such as using algebraic manipulation, factoring, or using other limit theorems such as the squeeze theorem. However, L'Hopital's Rule is often the most efficient and straightforward method for evaluating these types of limits.

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