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- Find the following limit:

We have:

- Find the following limit:

We have:

- Find the following limit:

Take out a factor of *x*^{2} from each square root and then replace *x* by
:

Alternatively we can rationalize the square root and L'Hopital is not used:

- Find the limit
.

We find
.

Then *y* = *e*^{z}.

We have:

So
*y* = *e*^{0} = 1 is the desired limit.
- L'Hopital's Problem: find the following limit:

We first consider the case that *a* > 0.

We can simplify by writing *x* = *at*; then each term in the numerator has a factor of *a*^{2} and each term in the denominator a factor of
*a*, giving a net factor of *a*.

Apart from this factor, the problem is the same as if *a* were put to the value one.

So we consider the limit:

So, if *a* > 0, the required limit is
.

If *a* = 0 the problem is ill-posed.

Finally if *a* is negative, we just evaluate, at *x* = *a*, without using L'Hopital, giving
limit zero, since the numerator is zero, but the denominator is
.

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** Up:** Cauchy's Mean Value Theorem
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*George A. J. Sparling*

*2001-01-11*