Evaluate the following limit : lim(x→3) (x - 3)/(√(x - 2) - √(4 - x))

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 3}\cfrac{\text x-3}{\sqrt{\text x-2}-\sqrt{4-\text x}} \)

lim(x→3) (x - 3)/(√(x - 2) - √(4 - x))

Answer 1

Given \(\lim\limits_{\text x \to 3}\cfrac{\text x-3}{\sqrt{\text x-2}-\sqrt{4-\text x}} \)

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 \(\lim\limits_{\text x \to 3}\cfrac{\text x-3}{\sqrt{\text x-2}-\sqrt{4-\text x}} \) = \(\cfrac{1+1}2\) = 1