Single Variable Calculus

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Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 2

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The given functions $\lim\limits_{x \to 1^-} f(x) = 3$ and $\lim\limits_{x \to 1^+} f(x) = 7$, explain what it means and state if its possible that $\lim\limits_{x \to 1}$ exists.

The meaning of these limits is that as $x$ approaches 1 from the negative side, the limit of the graph goes towards a $y$-value of 3. On the other hand, if we consider $x$ that approaches 1 from the positive side, the limit of the graph goes towards a $y$-value of 7.

It is not possible that $\lim\limits_{x \to 1}$ exists because as stated in the definition, $\lim\limits_{x \to a} f(x) = L$ if and only if $\lim\limits_{x \to a^-} f(x) = L$ and $\lim\limits_{x \to a^+} f(x) = L$. The limit of the function as $x$ approaches 1 does not exist because the values are different as $x$ approaches 1 from left and right.

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