As we need to find \(\lim\limits_{\text x \toπ/2}\cfrac{a^{cot \text x}-a^{cos\,\text x}}{cot \,\text x-cos\,\text x}\)
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 = \(\lim\limits_{\text x \toπ/2}\cfrac{a^{cot \text x}-a^{cos\,\text x}}{cot \,\text x-cos\,\text x}\)
\(=\cfrac00\)(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 \(\lim\limits_{\text x \to0}\cfrac{log(1+\text x)}{\text x}=1\)
This question is a direct application of limits formula of exponential limits.
{using properties of exponents}
As, x→ (π/2)
∴ cot(π/2) – cos(π/2) → 0
Let, y = cot x – cos x
∴ if x→π/2 ⇒ y→0
Hence, Z can be rewritten as-
Use the formula: \(\lim\limits_{\text x \to0}\cfrac{a^{\text x-1}}{\text x}\) = log a
\(\therefore\) Z = log a
Hence,
\(\lim\limits_{\text x \toπ/2}\cfrac{a^{cot \text x}-a^{cos\,\text x}}{cot \,\text x-cos\,\text x}\) = log a
