Question

Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why.

\(\begin{array}{l}(a)\mathop {lim}\limits_{x \to {2^ - }} f(x) (b)\mathop {lim}\limits_{x \to {2^ + }} f(x) (c) \mathop {lim}\limits_{x \to 2} f(x)\\(d) f(2) (e)\mathop {lim}\limits_{x \to 3} f(x) (f)f(4)\end{array}\)

Short answer

(a)\(\mathop {\lim }\limits_{x \to {2^ - }} f(x) = 3\)

(b)\(\mathop {\lim }\limits_{x \to {2^ - }} f(x) = 1\)

(c)\(\mathop {\lim }\limits_{x \to 2} f(x)\)doesn’t exist.

(d)\(f\left( 2 \right) = 3\)

(e) \(f\left( 4 \right)\)is not defined.

Step-by-step solution

Limit of a function

The limit of a function at a point exists if the value of the left-hand limit and right-hand limit is equal.

\(\mathop {\lim }\limits_{x \to a} f\left( x \right) = \mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right)\)

Step 2:\((a)\mathop {lim}\limits_{x \to {2^ - }} f(x)\)

In the given figure from the left side, the value \(\mathop {\lim }\limits_{x \to {2^ - }} f(x)\)is equal to 3 because the graph touches point 2 at 3 of the y axis.

Therefore,

\(\mathop {\lim }\limits_{x \to {2^ - }} f(x) = 3\)

Step 3:\((b)\mathop {lim}\limits_{x \to {2^ + }} f(x)\)

In the given figure from the right side, the value \(\mathop {\lim }\limits_{x \to {2^ - }} f(x)\)is equal to 1 because the graph touches point 2 at 1 of the y-axis.

Therefore,

\(\mathop {\lim }\limits_{x \to {2^ - }} f(x) = 1\)

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Step 4:\((c) \mathop {lim}\limits_{x \to 2} f(x)\)

The value of \(\mathop {\lim }\limits_{x \to {2^ - }} f(x)\)and \(\mathop {\lim }\limits_{x \to {2^ - }} f(x)\)is not equal, hence \(\mathop {\lim }\limits_{x \to 2} f(x)\)does not exist.

Step 5:\((d) f(2) \)

From the graph, it can be found that at\(x = 3\), the value of the function is 3.

Therefore,

\(f\left( 2 \right) = 3\)

Step 6:\((e)\mathop {lim}\limits_{x \to 3} f(x)\)

From the left and right sides, the value of \(\mathop {\lim }\limits_{x \to {3^ - }} f(x)\)and \(\mathop {\lim }\limits_{x \to {3^ + }} f(x)\)is 2. Both the values are equal hence the value of \(\mathop {\lim }\limits_{x \to 3} f(x)\)will be 2.

Therefore,

\(\mathop {\lim }\limits_{x \to 3} f(x) = 2\)

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Step 7:\((f)f(4)\)

The function is not defined at\(x = 4\).

Thus, \(f\left( 4 \right)\) doesn’t exist.

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