# Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{x}}{x^2}}$

The cosine of angle $x$ is subtracted from one and the difference of them is divided by the square of variable $x$. The quotient of them formed a rational function and its limit should be evaluated as the value of $x$ closer to $0$ in calculus.

$\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{x}}{x^2}}$

Let us try to find the limit of one minus cosine of angle $x$ by $x$ squared as $x$ approaches zero by using the direct substitution.

$=\,\,\,$ $\dfrac{1-\cos{(0)}}{(0)^2}$

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

$=\,\,\,$ $\dfrac{1-1}{0}$

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

It is calculated that the limit of one minus cosine of angle $x$ by $x$ squared as $x$ closer to $0$ is intermediate and it clears that the direct substitution method is not recommendable to find the limit.

## Methods

However, it can be evaluated in two different methods. So, let’s learn how to find the limit of the quotient of one minus cos of angle $x$ by square of $x$ as $x$ tends to $0$.

### Fundamental method

Learn how to calculate the limit of $1$ minus cos of angle $x$ by $x$ square as $x$ approaches $0$ by using the combination of both trigonometric identities and limit rules.

### L’Hopital’s | L’Hospital’s Rule

Learn how to find the limit of one minus cos of angle $x$ by square of $x$ as $x$ tends to $0$ by applying the L’Hopital’s (or L’Hospital’s) rule.

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