Let f(x)=(cos2-cos2 x)/(x^2-|x|), then (a) Limx →-1 f(x)=2 sin2 (b) Limx →1 f(x)=2 sin2 (c) Limx

step 1 in this problem note that since the function is continuous everywhere left and limit and the rig

Let f(x)=(cos2-cos2 x)/(x^2-|x|), then (a) Limx →-1 f(x)=2...

Let $f(x)=\frac{\cos 2-\cos 2 x}{x^{2}-|x|}$, then
(a) $\operatorname{Lim}_{x \rightarrow-1} f(x)=2 \sin 2$
(b) $\operatorname{Lim}_{x \rightarrow 1} f(x)=2 \sin 2$
(c) $\operatorname{Lim}_{x \rightarrow-1} f(x)=2 \cos 2$
(d) $\operatorname{Lim}_{x \rightarrow 1} f(x)=2 \cos 2$

Key Concepts

Limits of Functions

Limits are the foundation of calculus, describing the behavior of a function as its input approaches a particular point. They allow us to determine the value that a function is approaching even if it is not defined at that exact point, which is essential for understanding continuity, derivatives, and the overall behavior of functions near points of interest.

Indeterminate Forms and l'Hôpital's Rule

When direct substitution in a limit results in an indeterminate form such as 0/0 or ?/?, it indicates that further analysis is needed to evaluate the limit. l'Hôpital's Rule is a key tool in these situations, allowing one to differentiate the numerator and denominator separately and then take the limit, thereby resolving the indeterminacy.

Absolute Value Function and Piecewise Definitions

The absolute value function has a natural piecewise definition because it outputs different values depending on whether the input is positive or negative. In limit problems, especially those where the variable approaches a point that affects the sign of the input inside the absolute value, it is crucial to consider the function's behavior in each region defined by the absolute value.

Trigonometric Functions and Their Behavior

Trigonometric functions, such as cosine and sine, often appear in limit problems and have well-known behaviors and identities. Understanding how these functions behave near specific points, as well as applying trigonometric identities or small-angle approximations, can greatly facilitate the evaluation of limits involving trigonometric expressions.

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