Question About Finding Limits with L'Hopital's Rule

In summary, the student attempted to solve the homework problem but did not understand why the limit was allowed to be skipped and the derivative of arcsin(3x/4) was used instead.
  • #1
silvershine
13
0

Homework Statement


A question we had for homework was: Which of the following is equivalent to limh->0 [arcsin((3(x+h))/4) – arcsin (3x/4)]/h ?

Homework Equations


There were multiple answer choices, but the correct answer is 3/(√16-9x^2).

The Attempt at a Solution


We've already walked through the problem in class so I know the steps to solve it, but there's one step that I didn't understand and he couldn't explain quite clearly to me. Why are we allowed to just calculate the derivative of arcsin(3x/4)? I know that according to L'Hopital's Rule we can calculate the limit by finding the limit of the derivatives, but why can we skip the other parts of the function altogether?
 
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  • #2
It is not an application of L'Hopital's rule. The limit you are given is the definition of the derivative of f(u) = arcsin(3u/4) at the point x. Recall that the derivative of f(x) at the point a is equal to the limit of the difference quotient:
[tex] f'(a) = \lim_{h\rightarrow 0} \frac{f(a + h) - f(a)}{h}[/tex]
If you see a limit that is of the form on the right side of this equation, and you know the derivative f'(a) by another method, then there is no need to go through the rigmarole of algebraic limit deconstruction: just use your knowledge that it is equal to f'(a).
 
  • #3
So you would just be plugging 0 in for h and solving the limit normally?
 
  • #4
We cannot replace h by 0 directly, as division by 0 is undefined, and in any case would only be equal to the limit if the expression is continuous at h = 0. Not all functions are continuous or have continuous derivatives!
While it may be possible to simplify the limit by going through extended analytical means, the fact that the limit is equal to the derivative, and we already know the derivative by simpler methods, means we can go ahead and replace the entire limit by its equivalent: the derivative. You should review the definition of the derivative (not just the shortcuts for its calculation) and practice recognizing this type of limit.
 
  • #5
I'll definitely do that. Thank you!
 

FAQ: Question About Finding Limits with L'Hopital's Rule

What is L'Hopital's Rule?

L'Hopital's Rule is a mathematical concept used to find the limit of a function as it approaches a certain point. It states that if the limit of a function can be expressed as the ratio of two functions, and the limit of the numerator and denominator both exist, then the limit of the original function can be found by taking the limit of the ratio of the derivatives of the numerator and denominator.

When should L'Hopital's Rule be used?

L'Hopital's Rule should be used when the limit of a function cannot be evaluated using direct substitution, or when using other limit theorems such as the Squeeze Theorem or the Limit Laws is not practical or possible.

How do you use L'Hopital's Rule?

To use L'Hopital's Rule, first determine if the limit of the original function can be expressed as the ratio of two functions. Then, take the derivative of both the numerator and denominator, and evaluate the limit of the resulting ratio. If the limit of the ratio exists, then it is equal to the limit of the original function.

Are there any restrictions or limitations to using L'Hopital's Rule?

Yes, there are some restrictions to using L'Hopital's Rule. The functions in the original limit must be differentiable in a neighborhood of the point where the limit is being evaluated. Additionally, the limit of the ratio of the derivatives must exist and be finite. If these conditions are not met, then L'Hopital's Rule cannot be used to find the limit.

Can L'Hopital's Rule be used for all types of functions?

No, L'Hopital's Rule can only be used for certain types of functions. It can be used for functions that approach either 0, infinity, or negative infinity when evaluated at a certain point. It cannot be used for functions that approach a finite number, as the limit of the ratio of the derivatives would not exist in this case.

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