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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.


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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