Limits of even functions A function f is even if f(-x)=f(x)...

Limits of even functions A function $f$ is even if $f(-x)=f(x)$ for all $x$ in the domain of $f$. If $f$ is even, with $\lim _{x \rightarrow 2^{+}} f(x)=5$ and $\lim _{n \rightarrow \infty} f(x)=8 .$ find the following limits. $x \rightarrow 2$ a. $\lim _{x \rightarrow-2^{+}} f(x)$ b. $\lim _{x \rightarrow-2^{-}} f(x)$

Limits of even functions A function $f$ is even if $f(-x)=f(x)$ for all $x$ in the domain of $f$. If $f$ is even, with $\lim _{x \rightarrow 2^{+}} f(x)=5$ and $\lim _{n \rightarrow \infty} f(x)=8 .$ find the following limits.
$x \rightarrow 2$
a. $\lim _{x \rightarrow-2^{+}} f(x)$
b. $\lim _{x \rightarrow-2^{-}} f(x)$

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Key Concepts

Limit of a Function

The limit of a function at a point is a fundamental concept in calculus that describes what value the function approaches as the independent variable nears a specific point. Establishing the limits helps in understanding the continuity and local behavior of functions, and is especially useful when combined with symmetry properties in functions.

Even Functions

Even functions are characterized by the property f(-x) = f(x) for every x in the function's domain. This symmetry about the y-axis means that the behavior of the function on the negative side of the x-axis mirrors its behavior on the positive side, which is essential when analyzing the function’s limits at points that are negatives of each other.

One-Sided Limits

One-sided limits describe the behavior of a function as the input approaches a particular point from one side only—either from the left or the right. They are crucial in situations where the function may exhibit different behaviors depending on the direction of approach, allowing for a finer analysis of the function’s local behavior.

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