Question

use the given graph of the function f to find the following limits:

1. $lim_{x

ightarrow1}f(x)=$ dne help (limits)

1. $lim_{x

ightarrow1^{+}}f(x)=0$

1. $lim_{x

ightarrow1^{-}}f(x)=dne$

1. $lim_{x

ightarrow4}f(x)=0$

1. $f(4)=0$

Explanation:

Step1: Recall limit definition

The limit $\lim_{x
ightarrow a}f(x)$ exists if and only if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)$.

Step2: Analyze $\lim_{x

ightarrow 1^{-}}f(x)$
As $x$ approaches $1$ from the left - hand side, by looking at the graph, the function values approach a non - zero value.

Step3: Analyze $\lim_{x

ightarrow 1^{+}}f(x)$
As $x$ approaches $1$ from the right - hand side, the function values approach $0$. Since $\lim_{x
ightarrow 1^{-}}f(x)
eq\lim_{x
ightarrow 1^{+}}f(x)$, $\lim_{x
ightarrow 1}f(x)$ does not exist (DNE).

Step4: Analyze $\lim_{x

ightarrow 4^{-}}f(x)$ and $\lim_{x
ightarrow 4^{+}}f(x)$
As $x$ approaches $4$ from both the left - hand side and the right - hand side, the function values approach $0$. So $\lim_{x
ightarrow 4}f(x) = 0$. Also, from the graph, the value of the function at $x = 4$, $f(4)=0$.

Answer:

1. $\lim_{x

ightarrow 1}f(x)=DNE$

1. $\lim_{x

ightarrow 1^{+}}f(x)=0$

1. $\lim_{x

ightarrow 1}f(x)=DNE$

1. $\lim_{x

ightarrow 4}f(x)=0$

1. $f(4)=0$