# Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{mx}}{1-\cos{nx}}}$

The limit of the quotient of one minus cosine of angle $m$ times $x$ by one minus cos of angle $n$ times $x$ should has to evaluate as the value of $x$ approaches to zero in this problem.

$\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{mx}}{1-\cos{nx}}}$

Let’s try to find the limit of the rational function in trigonometric ratios by the direct substitution as the value of $x$ is closer to $0$.

$=\,\,\,$ $\dfrac{1-\cos{\big(m(0)\big)}}{1-\cos{\big(n(0)\big)}}$

$=\,\,\,$ $\dfrac{1-\cos{(m \times 0)}}{1-\cos{(n \times 0)}}$

$=\,\,\,$ $\dfrac{1-\cos{(0)}}{1-\cos{(0)}}$

According to the trigonometry, the cosine of angle zero radian is one.

$=\,\,\,$ $\dfrac{1-1}{1-1}$

$=\,\,\,$ $\dfrac{0}{0}$

The limit of the trigonometric function one minus cos of angle $m$ times $x$ by one minus cosine of $n$ times $x$ is indeterminate as the value of $x$ tends to zero. The indeterminate form clears that the direct substitution method is not recommendable to calculate the limit. However, it can be calculated in the following two different methods.

### Fundamental

Learn how to find the limit of $1$ minus cos of $mx$ by $1$ minus cosine of $nx$ as the value of $x$ is closer to $0$ fundamentally by the mathematical identities.

### L’Hospital’s Rule

Learn how to calculate the limit of $1$ minus cosine of angle $mx$ by $1$ minus cos of angle $nx$ as the value of $x$ approaches zero by the L’Hopital’s rule.

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