Evaluate the following limit : lim(x→3) (√(x + 3) - √6)/(x^2 - 9)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 3}\cfrac{\sqrt{\text x+3}-\sqrt 6}{\text x^2-9} \)

lim(x→3) (√(x + 3) - √6)/(x² - 9)

Answer 1

Given \(\lim\limits_{\text x \to 3}\cfrac{\sqrt{\text x+3}-\sqrt 6}{\text x^2-9} \)

To find: the limit of the given equation when x tends to 3

Substituting x as 3, we get an indeterminant form of \(\cfrac00\)

Rationalizing the given equation

Formula: (a + b) (a - b) = a² - b²

Now we can see that the indeterminant form is removed, so substituting x as 3

We get \(\lim\limits_{\text x \to 3}\cfrac{\sqrt{\text x+3}-\sqrt 6}{\text x^2-9} \) = \(\cfrac1{6(2\sqrt6)}=\cfrac1{12\sqrt6}\)