Evaluate the following limit lim(x→0) (e^x - 1)/(√(1 - cos x))

Question

Evaluate the following limit : \(\lim\limits_{\text x \to0}\cfrac{e^{\text x}-1}{\sqrt{1-cos\,\text x}} \)

lim(x→0) (e^x - 1)/(√(1 - cos x))

Answer 1

As we need to find \(\lim\limits_{\text x \to0}\cfrac{e^{\text x}-1}{\sqrt{1-cos\,\text x}} \) 

lim(x→0) (e^x - 1)/(√(1 - cos 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 = \(\lim\limits_{\text x \to0}\cfrac{e^{\text x}-1}{\sqrt{1-cos\,\text x}} \)

 \(=\cfrac00\)(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 \(\lim\limits_{\text x \to0}\cfrac{log(1+\text x)}{\text x}=1\)

It also involves a trigonometric term, so there is a possibility of application of Sandwich theorem-

\(\lim\limits_{\text x \to0} \cfrac{sin\,\text x}{\text x}=1\)

As Z = \(\lim\limits_{\text x \to0}\cfrac{e^{\text x}-1}{\sqrt{1-cos\,\text x}} \)

To apply the formula we need to get the form as present in the formula.

So we proceed as follows-

\(\because\) Z = \(\lim\limits_{\text x \to0}\cfrac{e^{\text x}-1}{\sqrt{1-cos\,\text x}} \)

Multiplying numerator and denominator by \(\sqrt{1+cos\,\text x}\)

{using algebra of limits}