The value of the limit is 0.
To evaluate the limit statement [tex]\(\lim_{x \to 0} f\left([g(x)]^2 + 1\right)\)[/tex] using the given graphs, do the following:
Evaluate (g(x) at x = 0:
Look at the graph of g(x) and find the value of g(x) when x = 0.
From the graph, when x = 0, g(0) is 1.
Calculate [tex]\([g(x)]^2\)[/tex] when x = 0:
Since g(0) = 1, [tex]\([g(0)]^2[/tex] = [tex]1^2[/tex] = 1.
Add 1 to [tex]\([g(x)]^2\)[/tex]:
[tex]\([g(0)]^2[/tex] + 1 = 1 + 1 = 2.
Evaluate f(2):
Look at the graph of f(x) to find the value of f(2).
From the graph, when x = 2, f(2) is 0.
Therefore, [tex]\(\lim_{x \to 0} f\left([g(x)]^2 + 1\right)[/tex] = f(2) = 0).
