Question

1. use the table to find

x	6.5	6.9	6.99	6.999	7	7.0001	7.001	7.01	7.1
f(x)	8.89	8.91	8.99	8.9999	dne	9.001	9.0118	9.023	9.045

a. \\(\lim_{x\to7^{-}}f(x)=\\) b) \\(\lim_{x\to7^{+}}f(x)=\\) c) \\(\lim_{x\to7}f(x)=\\) d) \\(f(7)=\\)

Explanation:

Step1: Find left - hand limit

As $x$ approaches $7$ from the left ($x\to7^-$), we look at values of $x$ less than $7$ in the table. As $x$ gets closer to $7$ (e.g., $x = 6.9,6.99,6.999$), $f(x)$ gets closer to $9$. So, $\lim_{x\to7^-}f(x)=9$.

Step2: Find right - hand limit

As $x$ approaches $7$ from the right ($x\to7^+$), we look at values of $x$ greater than $7$ in the table. As $x$ gets closer to $7$ (e.g., $x = 7.0001,7.001,7.01$), $f(x)$ gets closer to $9$. So, $\lim_{x\to7^+}f(x)=9$.

Step3: Find the overall limit

Since $\lim_{x\to7^-}f(x)=\lim_{x\to7^+}f(x) = 9$, then $\lim_{x\to7}f(x)=9$.

Step4: Find the function value at $x = 7$

From the table, $f(7)=\text{DNE}$ (does not exist).

Answer:

a. $\lim_{x\to7^-}f(x)=9$
b. $\lim_{x\to7^+}f(x)=9$
c. $\lim_{x\to7}f(x)=9$
d. $f(7)=\text{DNE}$