Question

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

Answer 1

\[\lim_{x \rightarrow 121}\frac{ \sqrt{x}-11 }{ x-121 }\]

Answer 2

can we factor the bottom?

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

do you mean at x=121?

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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?

Answer 13

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

Answer 14

yes

Answer 15

and since sqrt(x) = a

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

Answer 16

Thank you!

Answer 17

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