(i) `lim_(x to 0) (a^(x) - 1)/(log_(a)(1 + x)), a gt 0` (ii) `lim__(x to 0) (In (X + a)- In a)/(e^(2x) - 1)` (ii) `lim_(x to (pi)/(4)) (In(tanx))/(1 - cotx)`

Explanation

Let's solve the given limits step by step.

### (i) \( \lim_{x \to 0} \frac{a^x - 1}{\log_a(1 + x)} \)

1. **Convert the logarithm**: We know that \( \log_a(1 + x) = \frac{\ln(1 + x)}{\ln(a)} \). Therefore, we can rewrite the limit as:
\[ \lim_{x \to 0} \frac{a^x - 1}{\frac{\ln(1 + x)}{\ln(a)}} = \lim_{x \to 0} \frac{(a^x - 1) \ln(a)}{\ln(1 + x)} \]
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