Limits and Continuity of a Piecewise Function

by genevieveb
Tags: calculus, continuity, limit, piecewise function
 P: 5 1. The problem statement, all variables and given/known data Find a value for k to make f(x) continuous at 5 f(x)= sqrt(x2-16)-3/(x-5) if x cannot equal 5 3x+k when x=5 2. Relevant equations none 3. The attempt at a solution lim x->5 sqrt((x+4)(x-4))-3/(x-5) * sqrt((x+4)(x-4))+3/sqrt((x+4)(x-4))+3 lim x->5 (x+4)(x-4)-9/(x-5)[sqrt((x+4)(x-4))-3] And that's as far as I got. I'm not sure what my next step should be or if what I did is wrong. I graphed the function and used a program to solve for the limit which is apparently 5/3, but I couldn't come up with that answer. I would really appreciate any help/suggestions. Thanks.
 Emeritus Sci Advisor HW Helper Thanks PF Gold P: 11,002 You need to use parentheses to make it clear what you're doing. It looks like what you are doing is finding $$\lim_{x \to 5} \frac{\sqrt{(x+4)(x-4)}-3}{x-5}\cdot \frac{\sqrt{(x+4)(x-4)}+3}{\sqrt{(x+4)(x-4)}+3}$$ My first question to you is, why? I don't see why you have the x-5 in there. Look up the definition of continuity. You should see you're making this problem harder than intended. Or did you not describe f(x) correctly in the beginning of your post?
 P: 5 Sorry! I copied the question wrong the first time. I fixed it. It is all over (x-5) in the first part of the function. Sorry about that.
Emeritus