Evaluate the following limit : lim(x→0)(√(a^2+x^2) - a)/x^2

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 0}\cfrac{\sqrt{a^2+\text x^2}-a}{\text x^2} \)

lim(x→0)(√(a²+x²) - a)/x²

Answer 1

Given \(\lim\limits_{\text x \to 0}\cfrac{\sqrt{a^2+\text x^2}-a}{\text x^2} \) 

To find: the limit of the given equation when x tends to 0

Substituting x as 0, 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 0

We get, \(\lim\limits_{\text x \to 0}\cfrac{\sqrt{a^2+\text x^2}-a}{\text x^2} \) = \(\cfrac1{a+a}=\cfrac1{2a}\)