Question

graphical reasoning in exercises 29 and 30, use the graph of the function f to decide whether the value of the given quantity exists. if it does, find it. if not, explain why. 29. (a) f(1) (b) lim_{x→1}f(x) (c) f(4) (d) lim_{x→4}f(x)

Explanation:

Step1: Find $f(1)$

Locate $x = 1$ on the $x$-axis. The solid - dot at $x = 1$ has a $y$-value of $2$. So $f(1)=2$.

Step2: Find $\lim_{x

ightarrow1}f(x)$
As $x$ approaches $1$ from both the left - hand side and the right - hand side, the function values approach the $y$-value of the solid - dot at $x = 1$. So $\lim_{x
ightarrow1}f(x)=2$.

Step3: Find $f(4)$

Locate $x = 4$ on the $x$-axis. There is an open - dot at $x = 4$ and no solid - dot. So $f(4)$ does not exist because the function is not defined at $x = 4$.

Step4: Find $\lim_{x

ightarrow4}f(x)$
As $x$ approaches $4$ from both the left - hand side and the right - hand side, the function values approach the $y$-value of the open - dot. The $y$-value of the open - dot at $x = 4$ is $2$. So $\lim_{x
ightarrow4}f(x)=2$.

Answer:

(a) $f(1)=2$
(b) $\lim_{x
ightarrow1}f(x)=2$
(c) $f(4)$ does not exist
(d) $\lim_{x
ightarrow4}f(x)=2$