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.

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.