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Let $\lim_{x \to 2^+}(g(x))=5$ , $\lim_{x \to 2^-}(g(x))=-5$ . find $\lim_{x \to 2}(g(x))$ if it exists

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Determine end behavior of polynomial and rational functions

Full solution

Q. Let

\lim_{x \to 2^+}(g(x))=5

,

\lim_{x \to 2^-}(g(x))=-5

. find

\lim_{x \to 2}(g(x))

if it exists

Identify Limit Approach: To find the limit of $g(x)$ as $x$ approaches $2$ , we need to consider the limits from both the left and the right side of $2$ .
Limit from Right: The limit of $g(x)$ as $x$ approaches $2$ from the right ( $2+$ ) is given as $5$ .
Limit from Left: The limit of $g(x)$ as $x$ approaches $2$ from the left ( $2-$ ) is given as $-5$ .
Check Equality of Limits: For the limit of $g(x)$ as $x$ approaches $2$ to exist, the left-hand limit and the right-hand limit must be equal.
Limits Not Equal: Since the left-hand limit is $-5$ and the right-hand limit is $5$ , they are not equal.
Conclusion: Therefore, the limit of $g(x)$ as $x$ approaches $2$ does not exist.

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