Question

lim x=0 log(tanx)/logx​

Answer 1

Step-by-step explanation:

1 is the answer.

THANK YOU

Answer 2

Given,

f(x) = $\frac{log(tanx)}{log(x)}$

To find,

g(x) = $\lim_{x \to \ 0} f(x)$

Solution,

The value of g(x) is 1.

We can simply solve the mathematical question by the following procedure.

We are supposed to find the value of g(x).

g(x) = $\lim_{x \to \ 0} f(x)$

      = $\lim_{x \to \ 0} \frac{log(tanx)}{log(x)}$

We know by limits that,

∴  $\lim_{x \to \ 0} \frac{tanx}{x} = 1$

This can be proved by the L' Hospital rule;

We know that the current form of the function is $\frac{0}{0}$, thus we can apply the L' Hospital rule.

⇒ $\lim_{x \to \ 0} \frac{sec^2x}{1}$

⇒ 1

We can, now, substitute $\frac{tanx}{x}$ in the function f(x) to further calculate the value of g(x).

⇒ g(x) = $\lim_{x \to \ 0} \frac{log(\frac{tanx}{x}*x) }{log(x)}$

           = $\lim_{x \to \ 0} \frac{log(\frac{tanx}{x})+ logx}{logx}$

           = $\lim_{x \to \ 0} \frac{logx}{logx}$

           = 1

As a result, the value of g(x) is 1.