Question

For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why.

\(\begin{array}{l}(a)\mathop {lim}\limits_{x \to 1} f(x) (b)\mathop {lim}\limits_{x \to {3^ - }} f(x) (c) \mathop {lim}\limits_{x \to {3^ + }} f(x)\\(d)\mathop {lim}\limits_{x \to 3} f(x) (e)f(3)\end{array}\)

Short answer

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

(b) \(\mathop {\lim }\limits_{x \to {3^ - }} f(x) = 1\)

(c) \(\mathop {\lim }\limits_{x \to {3^ + }} f(x) = 4\)

(d)The value of \(\mathop {\lim }\limits_{x \to 3} f(x)\)does not exist.

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

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:\((a)\mathop {lim}\limits_{x \to 1} f(x) \)

In the given figure from the left side, the value of\(\mathop {\lim }\limits_{x \to {1^ - }} f(x)\)is 2 and from the right side, the value of \(\mathop {\lim }\limits_{x \to {1^ + }} f(x)\) is also 2. Both the values are equal so the value of \(\mathop {\lim }\limits_{x \to 1} f(x)\)will be equal to them that is 2.

Therefore, \(\mathop {\lim }\limits_{x \to 1} f(x) = 2\)

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

From the graph, it can be found that when\(x \to {3^ - }\)then the value of\(y\)tends to 1.

Therefore,

\(\mathop {\lim }\limits_{x \to {3^ - }} f(x) = 1\)

Step 4:\((c) \mathop {lim}\limits_{x \to {3^ + }} f(x)\)

In the given figure, when the value of \(x \to {3^ + }\) then the value of\(y\)tends to 4.

Therefore,

\(\mathop {\lim }\limits_{x \to {3^ + }} f(x) = 4\)

\(\)

Step 5:\((d)\mathop {lim}\limits_{x \to 3} f(x)\)

Since the value of \(\mathop {\lim }\limits_{x \to {3^ - }} f(x)\)and \(\mathop {\lim }\limits_{x \to {3^ + }} f(x)\)are not equal, thus the value of \(\mathop {\lim }\limits_{x \to 3} f(x)\)does not exist.

(e) Evaluate\(f\left( 3 \right)\) 

From the graph, it can be found that if\(x = 3\)then the value of\(y = 3\).

Thus, \(f\left( 3 \right) = 3\)