Evaluate the following limit : lim(x→0) (√(1 + x + x^2) - √(x + 1))/2x^2

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 0}\cfrac{\sqrt{1+\text x+\text x^2}-\sqrt{\text x+1}}{2\text x^2}\)

lim(x→0) (√(1 + x + x²) - √(x + 1))/2x²

Answer 1

Given \(\lim\limits_{\text x \to 0}\cfrac{\sqrt{1+\text x+\text x^2}-\sqrt{\text x+1}}{2\text x^2}\)

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 \(\lim\limits_{\text x \to 0}\cfrac{\sqrt{1+\text x+\text x^2}-\sqrt{\text x+1}}{2\text x^2}\) = \(\cfrac{1}{2(1+1)}=\cfrac14\)