Evaluate the following limit : lim(x→0) (tan mx)/(tan nx)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to 0}\cfrac{tan\,m\text x}{tan\,n\text x}\)

lim(x→0) (tan mx)/(tan nx)

Answer 1

To find: \(\lim\limits_{\text x \to 0}\cfrac{tan\,m\text x}{tan\,n\text x}\) 

\(\lim\limits_{\text x \to 0}\cfrac{tan\,m\text x}{tan\,n\text x}\)

Multiplying and Dividing by mx in numerator & Multiplying and Dividing by nx in the denominator:

As, x → 0 ⇒ mx → 0 & nx → 0

Now, put mx = y and nx = t

Hence, the value of \(\lim\limits_{\text x \to 0}\cfrac{tan\,m\text x}{tan\,n\text x}\) = \(\cfrac{m}n\)