Question

Any Help please #CALCULUS #LIMITS
Find lim(x->3)(√(x+1)-2)/(|-x^2-x+12|), can't solve it by myself

Answer 1

\[\LARGE \lim_{x \rightarrow 3} \frac{\sqrt{x+1}-2}{|-x^2-x+12|}\]
Better?

Answer 2

I am getting different left and right limits. So the limit x->3 does not exist.

Answer 3

Find the roots of | -x^2 - x + 12 |.
They are -4 and 3.
-x^2 - x + 12 is positive when x = 3- and is negative when x = 3+.
when x > 3+, | -x^2 - x + 12 | = x^2 + x - 12.
limx->3+ { sqrt(x+1) - 2 } / { x^2 + x - 12 } = ? (apply L'H).

Similar procedure for lim x->3-

Answer 4

Thanks everyone for your fast responds and sorry for my slow reply(the school internet broke down again)
The question is given by my lecturer, if it's non exist limit, may I know what should I give as an answer?

Answer 5

The limit does not exist because the left hand side limit is -1/28 and the right hand side limit is +1/28 and the two limits are not the same.

Answer 6

Thank you, but when I multiply by conjugate, I still hey 0 over 0 when solving the limit, am I wrong at any point? Or if you don't mind, can you shows me some steps?

Answer 7

I wanted to get rid of the absolute bars in the denominator. To do that I found the roots so I can determine in what intervals the function is positive/negative.
-x^2 - x + 12 = -(x^2 + x - 12) = -(x+4)(x-3)
The roots are -4 and +3.

-x^2 - x + 12 is negative in the intervals: (-infinity, -4) and (3, infinity)
-x^2 - x + 12 is positive in the interval: (-4, 3).

The point we are interested in is x = 3.
Slightly to the left of 3, -x^2 - x + 12 is poistive.
Slightly to the right of 3, -x^2 - x + 12 is negative.

We can use this information to get rid of the absolute bars in the proper intervals.

Answer 8

x = 3-, -x^2 - x + 12 is positive and therefore, |-x^2 - x + 12| = -x^2 - x + 12
x = 3+, -x^2 - x + 12 is negative and therefore, |-x^2 - x + 12| = x^2 + x - 12

Answer 9

So the original limit of x->3 is split into two parts: lim x->3- and lim x->3+.

lim x->3- { sqrt(x+1) - 2 } / { |-x^2 - x + 12| } =
lim x->3- { sqrt(x+1) - 2 } / { -x^2 - x + 12 }.
This is an indeterminate form 0/0. Therefore, we can apply L'H. Differentiate top and bottom separately: lim x->3- { 1/2 * (x+1)^(-1/2) } / { -2x - 1 } =
{ 1/2 * (4)^(-1/2) } / { -6 - 1 } = 1/2 * 1/2 * (-1/7) = -1/28.

Can you do the right side limit?

Answer 10

Thanks for being so kind to help, now I figure it out much better, I had spend few hours with it, thanks again!

Answer 11

You are welcome.