Another limit problem

[stinajeana]

Solve limit x-->-1 ((sqrt 12x+21)-3)/x+1

How would I start solving this? (I don't want the answer)

 

[daon2]

To start:

\(\displaystyle \dfrac{\sqrt{12x+21}-3}{x+1}=\dfrac{\sqrt{12x+21}-3}{x+1}\cdot \dfrac{\sqrt{12x+21}+3}{\sqrt{12x+21}+3}\)

 

[stinajeana]

Ahhhh, of course! thank you very much

 

[stinajeana]

To start:

\(\displaystyle \dfrac{\sqrt{12x+21}-3}{x+1}=\dfrac{\sqrt{12x+21}-3}{x+1}\cdot \dfrac{\sqrt{12x+21}+3}{\sqrt{12x+21}+3}\)

Would the answer be 2?

 

[daon2]

Would the answer be 2?

Yessiree

 

[Jason76]

I'm assuming FOIL was used on the top, while the bottom was simply multiplied (to get the final answer).

 

[Deleted member 4993]

I'm assuming FOIL was used on the top, while the bottom was simply multiplied (to get the final answer).

For numerator, we used (preferably for this type of problem) (a+b)(a-b) = a² - b²

We multiply top and bottom with the "radical conjugate" - so that this form comes out.

so,

\(\displaystyle (\sqrt{12x + 21} - 3)(\sqrt{12x + 21} + 3) \ = \ (\sqrt{12x + 21})^2 - (3)^2 \ = \ (12x+21) - 9 = 12(x+1)\)

so now the overall expression becomes:

\(\displaystyle \dfrac{12}{(\sqrt{12x + 21} + 3)}\)

and evaluate it at the limiting value of x = -1.