Thread: Limits - Solve By Manipulation

Okay, so I know you have to rationalize the limit in order to get it out of the indeterminate form, but I can't seem to do it for these two questions. Can someone help, start me off?

a) Lim as x -> 4 of (x^2-4)/(sqrtx - 2)

b) Lim as x -> 0 of [sqrt(3-x) - sqrt(x+3)] / x

Thanks,
Stefan

a)

This is not an indeterminate form. You'll find that the limit does not exist. Consider the left and right handed limits and you'll see they're not equal.

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b)

Notice that something cancels Then you can plug in without getting an indeterminate form.

Hi, thanks or the help, a quick question though. When we rationalize why do we use the opposite sign, for example, (b) had sqrt + sqrt and when it was rationalized it was sqrt - sqrt. What is the actual reason this is done, I know it cancels but what proof is there that that always works?

Well, it all boils down to the difference of squares formula:

In particular, if and are square roots, we would like to get rid of them and by multiplying by the conjugate (in this case, ) we get rid of them and we get an expression without square roots, i.e.