Sure, let's tackle this problem step by step:
### Part a: The value of the limit at [tex]\( c = -3 \)[/tex]
To find the value of the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( c \)[/tex], we use the definition of the function [tex]\( f(x) = x^2 \)[/tex].
The limit is calculated as follows:
[tex]\[
\lim_{x \to -3} f(x) = f(-3) = (-3)^2 = 9
\][/tex]
So, the value of the limit at [tex]\( c = -3 \)[/tex] is 9.
### Part b: The largest open interval about [tex]\( c \)[/tex] on which [tex]\( |f(x) - \text{limit value}| < \epsilon \)[/tex]
We need to find the largest open interval around [tex]\( c = -3 \)[/tex] where the inequality [tex]\( |f(x) - 9| < 0.4 \)[/tex] holds true.
First, express the inequality:
[tex]\[
|x^2 - 9| < 0.4
\][/tex]
Simplify the inequality:
[tex]\[
-0.4 < x^2 - 9 < 0.4
\][/tex]
[tex]\[
8.6 < x^2 < 9.4
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
\sqrt{8.6} < |x| < \sqrt{9.4}
\][/tex]
Calculate the approximate values of the square roots:
[tex]\[
\sqrt{8.6} \approx 2.93
\][/tex]
[tex]\[
\sqrt{9.4} \approx 3.07
\][/tex]
Thus we have:
[tex]\[
2.93 < |x| < 3.07
\][/tex]
Since [tex]\( x \)[/tex] can be in the range from [tex]\( -3.07 \)[/tex] to [tex]\( -2.93 \)[/tex] and from [tex]\( 2.93 \)[/tex] to [tex]\( 3.07 \)[/tex]:
Focusing on the interval about [tex]\( c = -3 \)[/tex], we get:
[tex]\[
-3.07 < x < -2.93
\][/tex]
Rounded to one decimal place and represented in interval notation, the largest open interval about [tex]\( c = -3 \)[/tex] is:
[tex]\[
(-3.4, -2.6)
\][/tex]
So, the largest open interval about [tex]\( c = -3 \)[/tex] where [tex]\( |x^2 - 9| < 0.4 \)[/tex] is [tex]\((-3.4, -2.6)\)[/tex].
