calculus qeustion --- desperately need help?

Question

If f(x) = sqrt(2x + 5) - sqrt(x+7) / x-2 , every except x=2
and f(2)=k

and f is a continuous function at x=2, then k is equal to what?
Please show how you get the answer. THANKS a bunch!

Answer 1

For the love of all that is holy, is it really so hard to type in two pairs of parentheses? I'm really getting tired of having to guess what you people mean through esp. USE THE PROPER ORDER OF OPERATIONS, DAMNIT!



In order to make f a continuous function, it is necessary to make f(2) = [x→2]lim f(x). Therefore, we must find this quantity.

[x→2]lim (√(2x+5) - √(x+7))/(x-2)
Direct substitution yields (√9-√9)/0 → 0/0, so we use L'hopital's rule:
[x→2]lim (1/√(2x+5) - 1/(2√(x+7)))/1
Direct evaluation now yields:
1/√9 - 1/(2√9)
1/3 - 1/6
1/6

So k = [x→2]lim (√(2x+5) - √(x+7))/(x-2) = 1/6

Answer 2

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Answer 3

take the limit as x goes to 2 from the left and the right

Answer 4

If f(2) = k, then k is undefined.