limit of integral

[pistol]

can you help me with this limit

 

[DrPhil]

can you help me with this limit

We need to see your work to see where you are getting stuck.

What have you tried? Do you know l'Hospital's rule?

 

[pistol]

i know it, but don't know exactly what to do with the integral

 

[DrPhil]

i know it, but don't know exactly what to do with the integral

Aha. According to the Fundamental Theorem of Calculus, what is the derivative with respect to the upper limit of a definite integral?

 

[pistol]

o yeah the integral of f(t) is f(x)... thanks

 

[pistol]

so with the L'Hopital's Rule i have lim -2cos(1/x^2)/x^3 as x--> 0, so it becomes undefined in (-∞;+∞)

 

[DrPhil]

so with the L'Hopital's Rule i have lim -2cos(1/x^2)/x^3 as x--> 0, so it becomes undefined in (-∞;+∞)

Hmmmm. We had better look at how l'Hospital's rule is applied. Since the limit of numerator/denominator is 0/0, an indeterminate form, we replace each by its derivative:

The numerator is \(\displaystyle \int_0^x \sin\frac{1}{t^2}dt\)
and its derivative is \(\displaystyle \sin\frac{1}{x^2}\)

The denominator is \(\displaystyle x\)
and its derivative is \(\displaystyle 1\)

The ratio of the derivatives is \(\displaystyle \sin\frac{1}{x^2}\)

 

[pistol]

aha thanks, so the function has no lmit when x-->0?