Calculation of limit. L'Hopital's rule

In summary, the problem is that the limit should be shown in a calculus textbook, and it can be obtained using L'Hopital's rule.
  • #1
EEristavi
108
5
Problem: Evaluate lim(x->0) x cotx

My attempt:
lim(x->0) x cotx = lim(x->0) x cosx / sinx = lim(x->0) cosx * lim(x->0) x / sinx = 1 * lim(x->0) x / sinx = lim(x->0) x / sinx

P.S.
I know I must/can use L'Hopital's rule to evaluate indeterminate limits, but no matter how many times I derive x/sinx I will always have sinx (in some power) in denominator.

I also tried different grouping of variables but still same scenario.

maybe I don't see something so even little hint would be nice...
 
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  • #2
EEristavi said:
Problem: Evaluate lim(x->0) x cotx

My attempt:
lim(x->0) x cotx = lim(x->0) x cosx / sinx = lim(x->0) cosx * lim(x->0) x / sinx = 1 * lim(x->0) x / sinx = lim(x->0) x / sinx
##\lim_{x \to 0} \frac{\sin(x)}x## is a well-known limit that should be shown in your calculus textbook. It's also a limit that can be obtained using L'Hopital.
EEristavi said:
P.S.
I know I must/can use L'Hopital's rule to evaluate indeterminate limits, but no matter how many times I derive x/sinx I will always have sinx (in some power) in denominator.
L'Hopital's Rule doesn't apply to all indeterminate limits, just those of the forms ##[\frac 0 0]## or ##[\pm \frac \infty \infty]##. Even then, it sometimes doesn't work, as it just gets you back to the same limit you started with.
EEristavi said:
I also tried different grouping of variables but still same scenario.

maybe I don't see something so even little hint would be nice...

BTW, in future posts, please don't delete the Homework Template.
 
Last edited:
  • #3
L'Hopital's rule says ##\lim_{x -> a} \frac{f(x)}{g(x)} = \lim_{x -> a} \frac{f'(x)}{g'(x)}## if both ##f(x)## and ##g(x)## tend either to ##0## or ##\infty## as ##x -> a##. Maybe you're using the quotient rule for derivatives which is wrong, because that's not what the rule says.
 
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  • #4
dgambh thank you very much. I was stuck on this for days and now I know why :D thank you very much again!

Mark44 thank you for too.
 

1. What is the purpose of calculating limits?

The purpose of calculating limits is to determine the behavior of a function as the input values approach a particular value, typically at infinity or a point of discontinuity. This can help in understanding the overall behavior of the function and making predictions about its values.

2. What is L'Hopital's rule and when is it used?

L'Hopital's rule is a mathematical technique used to evaluate limits of indeterminate forms, such as 0/0 or infinity/infinity. It states that if the limit of the ratio of two functions is indeterminate, then the limit of the ratio of their derivatives is equal to the original limit.

3. How do you apply L'Hopital's rule to solve a limit?

To apply L'Hopital's rule, first simplify the given limit to an indeterminate form. Then, take the derivative of both the numerator and denominator of the simplified limit. Finally, evaluate the limit using the derivative expressions.

4. Are there any restrictions on using L'Hopital's rule?

Yes, L'Hopital's rule can only be used when the limit is in an indeterminate form. It cannot be used to evaluate limits that are not in an indeterminate form, such as a constant or a finite number.

5. Can L'Hopital's rule be used to solve all limits?

No, L'Hopital's rule can only be applied to specific types of limits, such as indeterminate forms. There are other techniques, such as substitution and factoring, that can be used to solve limits that are not in an indeterminate form.

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