Using L'Hospital's Rule for Solving Limits: What Are the Steps?

  • Context: High School 
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Discussion Overview

The discussion revolves around the application of L'Hospital's Rule for solving limits, specifically focusing on the steps involved in using the rule and the conditions under which it can be applied. The conversation includes both theoretical explanations and practical examples related to limits in calculus.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant seeks guidance on the steps to follow when using L'Hospital's Rule for limits.
  • Another participant explains that if the limit of f(x)/g(x) is in an indeterminate form (0/0 or ∞/∞), one can instead consider the limit of f'(x)/g'(x), noting that this may need to be repeated.
  • A further contribution suggests that L'Hospital's Rule can also be applied in cases where one function approaches 0 and the other approaches infinity, by manipulating the expression into a suitable form.
  • There is a clarification that if a limit does not yield an indeterminate form, L'Hospital's Rule is not necessary, and the original limit should be used instead.

Areas of Agreement / Disagreement

Participants generally agree on the conditions under which L'Hospital's Rule can be applied, but there is some confusion regarding the necessity of manipulating functions to achieve an indeterminate form. This aspect remains somewhat contested, as one participant emphasizes that manipulation is not required if an indeterminate form is not present.

Contextual Notes

Some assumptions about the functions involved and their behavior near the limit point are not explicitly stated, which may affect the application of L'Hospital's Rule. Additionally, the discussion does not resolve the nuances of when manipulation is appropriate.

KAS90
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Hey there..
I studied limits long time ago ofcourse.. but I used to use an old way in solving them, because the l'hospital rule wasn't allowed:)
My question is..can someone please help in giving me the steps I should follow in solving a limit using L'hospital's rule?
Thanks a lot..I really need to know how to solve limits..
 
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If you're looking at the limit of f(x)/g(x), and it's in indeterminate form (either f and g go to 0, or f and g go to infinity) you can look at the limit of f'(x)/g'(x) instead, and if that limit exists it equals the limit of f/g. Sometimes you have to do this more than once
 
If you have a a fraction of the form f(x)/g(x) and f and g separately both go to 0 or both go to infinity (as x goes to a), then
[tex]\lim_{x\rightarrow a} \frac{f(x)}{g(x)}= \lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}[/tex]

That's what OfficeShredder said. I want to add that L'Hopital's rule can be used in other cases:
If we have f(x)g(x) with one of f(x) or g(x) going to 0 and the other to plus or minus infinity, then we can rewrite the problem as either f(x)/(1/g(x)) or g(x)/(1/f(x)) so we have the "0/0" or "[itex]\infty/\infty[/itex]" case.

If we have F(x)= f(x)g(x) and f(x) and g(x) both go to 0, then we can take the logarithm: ln(F(x))= g(x)ln(f(x)). Now g(x) goes to 0 while ln(f(x)) goes to negative infinity, the previous case. If this new limit is A, then the limit of F is eA.
If we have f(x)g(x), with f and g both going to
 
Thanx a lot Office_shredder and hallsofIvy..
u guys were a lot of help..so u mean that if I don't get an indeterminate quantity, I should manipulate the functions to get an indeteminate quantity..aha..
Thanx a lot again...
 
No! That means if you don't get an indeterminate quantity, you don't NEED l'hospital's rule! Just use the quantity you did get.
 
oh ok..
I get it now..
Thanx again..
 

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