Evaluate the following limit : lim(x→1) (√(5x - 4) - √x)/(x^2 - 1)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 1}\cfrac{\sqrt{5\text x-4}-\sqrt {\text x}}{\text x^2 - 1} \)

lim(x→1) (√(5x - 4) - √x)/(x² - 1)

Answer 1

Given \(\lim\limits_{\text x \to 1}\cfrac{\sqrt{5\text x-4}-\sqrt {\text x}}{\text x^2 - 1} \)

To find: the limit of the given equation when x tends to 1

Substituting x as 1 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 1

We get \(\lim\limits_{\text x \to 1}\cfrac{\sqrt{5\text x-4}-\sqrt {\text x}}{\text x^2 - 1} \) = \(\cfrac{4}{2(2)}\) = 1