Question

use the graph of the function f to decide whether each quantity exists. (if an answer does not exist, enter dne.)
(a) $f(1)$
(b) $\lim_{x \to 1} f(x)$
(c) $f(5)$
(d) $\lim_{x \to 5} f(x)$

Explanation:

Step1: Evaluate $f(1)$

Locate $x=1$ on the graph; the filled dot gives $f(1)=4$.

Step2: Evaluate $\lim_{x \to 1} f(x)$

Check left/right limits at $x=1$: both approach 6.
$\lim_{x \to 1^-} f(x) = 6$, $\lim_{x \to 1^+} f(x) = 6$, so $\lim_{x \to 1} f(x)=6$.

Step3: Evaluate $f(5)$

Locate $x=5$ on the graph; there is no filled dot, so $f(5)$ does not exist.

Step4: Evaluate $\lim_{x \to 5} f(x)$

Check left/right limits at $x=5$: both approach 4.
$\lim_{x \to 5^-} f(x) = 4$, $\lim_{x \to 5^+} f(x) = 4$, so $\lim_{x \to 5} f(x)=4$.

Answer:

(a) $4$
(b) $6$
(c) $\text{DNE}$
(d) $4$