## clara1223 one year ago find the limit as x approaches 121 of (sqrt(x)-11)/(x-121). I have already solved and gotten 1/22 which was right but I can't remember how and can't find my notes. All I need is an explanation

1. clara1223

$\lim_{x \rightarrow 121}\frac{ \sqrt{x}-11 }{ x-121 }$

2. amistre64

can we factor the bottom?

3. clara1223

if x was squared then yes but it's not.

4. amistre64

oh but it is ... let a^2 = x, then a = sqrt(x)

5. clara1223

wait I don't understand that, can you explain it further?

6. amistre64

also, as a general rule, if the top and bottom have the same zero ... the zeros cancel out due to a common factor.

7. amistre64

at x=11, we have 0/0 right?

8. clara1223

do you mean at x=121?

9. amistre64

its just a substitution ... if the x does not look like its a square ... replace it by something that looks squarey yeah, x = 121 ... brain is faster then my fingers sometimes

10. clara1223

how would you go about replacing it with something that looks "squarey"?

11. amistre64

if we let x=a^2, and take the sqrt of each side ... sqrt(x) = a a-11 --------- a^2 - 121 now it looks squared to me

12. clara1223

Oh! that helps a lot. So then can factor the denominator to (a+11)(a-11) and cancel a-11 from both sides so im left with 1/a+11?

13. amistre64

your notes might say; divide it al by the highest power of x tho

14. amistre64

yes

15. amistre64

and since sqrt(x) = a sqrt(121) = 11 and we get your coveted 1/22

16. clara1223

Thank you!

17. amistre64

factoring seems to be the best route to me ... and youre welcome again :)