Calc 1 :limits of quotients

lim
x-->-3
2- the square root of x^2-5 / x+3

can anyone walk me through this?

Lim x--> -3 of [2 - (x^2 - 5)^(1/2)] / [x + 3]

Multiply the function by [2 + (x^2 - 5)^(1/2)] / [2 + (x^2 - 5)^(1/2)]

Note that in the numerator we will have:

[2 - (x^2 - 5)^(1/2)][2 + (x^2 - 5)^(1/2)] = 4 - [x^2 - 5] = -x^2 + 9 = -(x + 3)(x - 3)

Lim x--> -3 of [-(x + 3)(x - 3)] / { [x + 3][2 + (x^2 - 5)^(1/2)] }

Now the two factors of (x + 3) will cancel:

Lim x--> -3 of [-(x - 3)] / [2 + (x^2 - 5)^(1/2)]

Now you can just sub in -3

[-(-3 - 3)] / [2 + ((-3)^2 - 5)^(1/2)]
= 3/2

Thus the limit is 3/2.

Done!