Evaluate the following limit : lim(x→2)(√(3- x)- 1)/(2 -x)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 2}\cfrac{\sqrt{3-\text x}-1}{2-\text x}\)

lim(x→2)(√(3- x)- 1)/(2 -x)

Answer 1

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

To find: the limit of the given equation when x tends to 2

Substituting x as 2, 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 2

We get \(\lim\limits_{\text x \to 2}\cfrac{\sqrt{3-\text x}-1}{2-\text x}\) =\(\lim\limits_{\text x \to 2}\cfrac{1}{\sqrt{3-\text x}+1}\) = \(\cfrac1{1+1}=\cfrac12\)