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

Question

Evaluate the following limit : \(\lim\limits_{\text x \to0}\cfrac{\sqrt{1+\text x}-1}{log(1+\text x)} \)

lim(x→0) (√(1 + x) - 1)/(log(1 + x))

Answer 1

As we need to find \(\lim\limits_{\text x \to0}\cfrac{\sqrt{1+\text x}-1}{log(1+\text x)} \)

lim(x→0) (√(1 + x) - 1)/(log(1 + x))

We can directly find the limiting value of a function by putting the value of the variable at which the limiting value is asked if it does not take any indeterminate form (0/0 or ∞/∞ or ∞-∞, .. etc.)

Let Z = \(\lim\limits_{\text x \to0}\cfrac{\sqrt{1+\text x}-1}{log(1+\text x)} \)

 \(=\cfrac00\)(indeterminate)

∴ We need to take steps to remove this form so that we can get a finite value.

TIP: Most of the problems of logarithmic and exponential limits are solved using the formula

   \(\lim\limits_{\text x \to0}\cfrac{a^{\text x-1}}{\text x}\) = log a and \(\lim\limits_{\text x \to0}\cfrac{log(1+\text x)}{\text x}=1\)

As Z = \(\lim\limits_{\text x \to0}\cfrac{\sqrt{1+\text x}-1}{log(1+\text x)} \)

To apply the formula of logarithmic limits we need to get the form that matches with one in formula

∴ multiplying numerator and denominator by \(\sqrt{1+\text x}+1\)

{using (a + b)(a - b) = a²- b²}

Use the formula: \(\lim\limits_{\text x \to0}\cfrac{log(1+\text x)}{\text x}=1\)

\(\therefore\) Z = 1/2

Hence,