Help/advice on Calculus I (Limits)?

Question

I have these two problems that I can't seem to get the hang of.

The first is as follows:

the limit of x approaches 3 of (sqrt x - sqrt 3) / (x - 3)

I realize that substituting a 3 makes both the numerator and denominator zero out, so there must be a common factor.

I tried multiplying both sides by the conjugate ( sqrt x + sqrt 3 ).

Maybe my math is wrong, but I come up with ( x - 3^2 ) / (x - 3)(sqrt x + sqrt 3)

I then tried to "foil" out the denominator, but came up with nothing I could use.

The other problem:

The limit as x approaches "a" from the positive side of

abs( x - a ) / ( x - a )

I know that the top number will be a positive number the entire time, but the bottom number could be positive OR negative. My guess is that the limit would not exist, but I'm not sure.

Help would be appreciated.

Answer 1

You're right, your math is wrong.

(√x - √3)(√x + √3) = (√x)² - (√3)² = x - 3

|x-a| = x-a if x-a > 0, that is, if x > a. But if x approaches a from the positive side, then x > a so

lim |x-a|/(x-a) =

x→a+

lim (x-a)/(x-a) = 1

x→a+