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Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1-\cos{x}}{x^2}}$ by L’Hopital’s or L’Hospital’s Rule

The limit of one minus cosine of angle x by x square is evaluated when the value of x closer to 0 as per the trigonometric identities and limit rules.

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

According to the direct substitution, the limit of one minus cos of angle x by square of x as x approaches 0 is indeterminate. Hence, the limit of the rational function can be calculated by using L’Hopital’s rule (or L’Hospital’s Rule).

Use the L’Hopital’s Rule

Differentiate both numerator and denominator with respect to $x$ to apply the L’Hospital’s Rule.

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}{(1-\cos{x})}}{\dfrac{d}{dx}{(x^2)}}}$

The derivative of difference of the terms can be evaluated by using the subtraction rule of the derivatives.

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}{(1)}-\dfrac{d}{dx}{(\cos{x})}}{\dfrac{d}{dx}{(x^2)}}}$

Find the derivative of one with respect to $x$ by using the derivative rule of a constant, calculate the differentiation of the cosine function with respect to $x$ as per derivative rule of cosine and also evaluate the derivative of square of $x$ by using the power rule of derivatives.

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{0-(-\sin{x})}{2x}}$

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sin{x}}{2x}}$

Use the L’Hospital’s Rule

The limit of the rational function becomes indeterminate as the value of $x$ approaches zero. So, let’s try L’Hopital’s rule one more time by differentiating both numerator and denominator with respect to $x$.

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}{(\sin{x})}}{\dfrac{d}{dx}{(2x)}}}$

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}{(\sin{x})}}{\dfrac{d}{dx}{(2 \times x)}}}$

The constant factor can be excluded from the differentiation as per the constant multiple rule of the derivatives.

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\dfrac{d}{dx}{(\sin{x})}}{2 \times \dfrac{d}{dx}{(x)}}}$

Now, find the differentiation of the sine function as per the derivative rule of sine function and also find the derivative of the variable with respect to same variable as per the derivative rule of the variable.

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

$=\,\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\cos{x}}{2}}$

Evaluate the Limit of the function

It is right time to find the limit of the cosine of angle $x$ by $2$ as $x$ closer to $0$ as per the direct substitution method.

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

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

$=\,\,\,$ $\dfrac{1}{2}$

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