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Evaluate the left hand and right hand limits of the following function at x = 2. Does \(\lim\limits_{x \to 2}f(x)\) exist?

\( f(x) = \begin{cases} 2x+3 & \quad \text{if } x \leq{2}\\ x+5, & \quad \text{if } x>{ 2} \end{cases} \)

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L.H.L,

\(\lim\limits_{x \to 2^-}f(x) \)

\(\lim\limits_{x \to 2^-}(2x+3) \)

= 2 + 5

= 7

And R.H.L,

= \(\lim\limits_{x \to 2^+}f(x)\lim\limits_{x \to 2^+}(x+5)\)

= 2 + 5

= 7

Since \(\lim\limits_{x \to 2^-}f(x) \) = \(\lim\limits_{x \to 2^+}f(x) \)=7

∴ \(\lim\limits_{x \to 2^+}f(x) \) exists and is equal to 7.

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