Question

Integral Help!

Answer 1

\[\lim_{x \rightarrow 1}\left( \frac{ \int\limits_{1}^{x}e ^{t ^{2}}dt }{ x^{2}-1 } \right)\]

Answer 2

As x approaches 1, the integral approaches 0, and so does the denominator. Have you tried L'Hopital's rule?

Answer 3

I'm not familiar with L'Hopital's rule, and the answer choices are 0, 1, e/2, 3, and nonexistent

Answer 4

@SithsAndGiggles what is your thought process behind this problem?

Answer 5

You have learned integrals but don't know L'Hopital's rule? Very suspicious.

Answer 6

Well, by evaluating directly, we see that
\[\lim_{x \rightarrow 1}\frac{ \int\limits_{1}^{x} e^{t^2}dt}{ x^2-1 }=\frac{ \int\limits_{1}^{1} e^{t^2}dt}{ 1^2-1 }=\frac{0}{0},\]
which is indeterminate form, and hence my wanting to use L'Hopital's rule.

Answer 7

Which is where you take the derivative of the numerator and denominator individually, then find the limit of the ratio of their derivatives. Sure you haven't done that before?

Answer 8

We skipped that chapter, but Khan (from Khan Academy) did just nicely explain it.

Answer 9

First time I've heard a calculus course skip the section involving L'Hopital's rule. In any case, do you get the gist of the rule?

Answer 10

Yes I do

Answer 11

Okay, so what are the derivatives of the numerator and denominator? For the numerator, use the Fundamental Theorem of Calculus. (Which I hope your class didn't also skip.)

Answer 12

Would the derivative of the numerator include the integral of e^(t^2) from 1 to x, or would it simply be the derivative of e^(t^2)?

Answer 13

It would include the integral:
\[\frac{d}{dx}\left[\int_1^x e^{t^2}dt\right]\]
The first (or second, depending on which you learn first) part of the FTC should show how to differentiate.

Answer 14

The derivative of the numerator would then be the original function, which is 2te^(t^2), correct?

Answer 15

2te^(t^2) - 2e

Answer 16

No, that's not it. The part of the FTC I'm referring to says that
\[\frac{d}{dx}\left[\int_c^{u(x)}f(t)dt\right] =f(u(x)) \cdot u'(x), \]
where u(x) is some function of x and c is a constant.

Answer 17

\[\frac{ d }{ dt }\left( \int\limits_{1}^{x}e ^{t ^{2}}dt \right) = 0\]
\[\frac{ d }{ dx }x ^{2}-1=2x\]

Answer 18

You're differentiating with respect to X, not T, in both numerator and denominator.

Answer 19

So e^(x^2)?

Answer 20

Yes, that's it.

Answer 21

yeah, it's no longer an indeterminate form.

Answer 22

Sith have you looked at my question?

Answer 23

e/2

Answer 24

That's correct.