Determine limx →0{(sinx-x)/(x^2)} Substituting x=0 gives limx →0{(sinx-x)/(x^2)}=(sin0-0)/(0)=(0

step 1 equal 0 so this becomes sign 0 minus 0 upon 0 square equal 0 by 0. So now apply L hospital rule.

Determine limx →0{(sinx-x)/(x^2)} Substituting x=0 gives...

Determine $\lim _{x \rightarrow 0}\left\{\frac{\sin x-x}{x^{2}}\right\}$ Substituting $x=0$ gives $$ \lim _{x \rightarrow 0}\left\{\frac{\sin x-x}{x^{2}}\right\}=\frac{\sin 0-0}{0}=\frac{0}{0} $$ Applying L'Hôpital's rule gives $$ \lim _{x \rightarrow 0}\left\{\frac{\sin x-x}{x^{2}}\right\}=\lim _{x \rightarrow 0}\left\{\frac{\cos x-1}{2 x}\right\} $$ Substituting $x=0$ gives $$ \frac{\cos 0-1}{0}=\frac{1-1}{0}=\frac{0}{0} \text { again } $$ Applying L'Hôpital's rule again gives $$ \lim _{x \rightarrow 0}\left\{\frac{\cos x-1}{2 x}\right\}=\lim _{x \rightarrow 0}\left\{\frac{-\sin x}{2}\right\}=0 $$

Determine $\lim _{x \rightarrow 0}\left\{\frac{\sin x-x}{x^{2}}\right\}$
Substituting $x=0$ gives
$$
\lim _{x \rightarrow 0}\left\{\frac{\sin x-x}{x^{2}}\right\}=\frac{\sin 0-0}{0}=\frac{0}{0}
$$
Applying L'Hôpital's rule gives
$$
\lim _{x \rightarrow 0}\left\{\frac{\sin x-x}{x^{2}}\right\}=\lim _{x \rightarrow 0}\left\{\frac{\cos x-1}{2 x}\right\}
$$
Substituting $x=0$ gives
$$
\frac{\cos 0-1}{0}=\frac{1-1}{0}=\frac{0}{0} \text { again }
$$
Applying L'Hôpital's rule again gives
$$
\lim _{x \rightarrow 0}\left\{\frac{\cos x-1}{2 x}\right\}=\lim _{x \rightarrow 0}\left\{\frac{-\sin x}{2}\right\}=0
$$

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative, which represents the rate of change of a function with respect to its variable. This concept is central to calculus and is particularly important when applying techniques like L'Hôpital's rule, as it relies on the derivatives of the functions involved.

L'Hôpital's Rule

L'Hôpital's rule is a strategy used in calculus to resolve limits that yield indeterminate forms. By differentiating the numerator and denominator separately, this rule often simplifies the limit evaluation process, making it possible to find the limit of a ratio of functions that initially appears undefined.

Limits

Limits are a foundational concept in calculus that examine the behavior of a function as its input approaches a particular value. They are essential for understanding continuity, derivatives, and integrals, providing insights into how functions behave near points of interest, including points where a function may not be explicitly defined.

Indeterminate Forms

Indeterminate forms occur when the direct substitution in a limit leads to ambiguous expressions like 0/0 or ?/?. Recognizing these forms is critical because they signal the need for more sophisticated techniques—such as L'Hôpital's rule—to evaluate the limit correctly.

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