As we need to find \(\lim\limits_{\text x \to2}\cfrac{\text x-2}{log_a(\text x-1)} \)
lim(x→2) (x - 2)/(loga(x - 1))
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 = \(\lim\limits_{\text x \to2}\cfrac{\text x-2}{log_a(\text x-1)}
\)
\(=\cfrac00\) (indeterminate form)
∴ 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 and logarithmic limits.
To get similar forms as in a formula, we move as follows-
As x→2 ∴ x - 2 →0
Let x - 2 = y
We can’t use the formula directly as the base of log is we need to change this to e.
Applying the formula for change of base-
We have-
Use the formula: \(\lim\limits_{\text x \to0}\cfrac{log(1+\text x)}{\text x}=1\)
\(\therefore\) Z = logea = log a
Hence,
\(\lim\limits_{\text x \to2}\cfrac{\text x-2}{log_a(\text x-1)} \) = log a
