Question

Let f be a function defined by (function shown in image)
Show that f is continuous at x=2

Answer 1

Using the continuity concept, since the lateral limits are equal as the numeric value of the function, the function is continuous at x = 2.

When a function f(x) is continuous at x = a?

It is continuous if the lateral limits are the same, and equal to the numeric value of the function, that is:

$\lim_{x \rightarrow a^-} f(x) = \lim_{x \rightarrow a^+} f(x) = f(a)$

In this problem, considering the two definitions, we have that:

• $\lim_{x \rightarrow 2^-} f(x) = \lim_{x \rightarrow 2} 4x - 7 = 4(2) - 7 = 8 - 7 = 1$.

• $\lim_{x \rightarrow 2^+} f(x) = \lim_{x \rightarrow 2} e^{x - 2} = e^{2 - 2} = e^0 = 1$.

• f(0) = 4(2) - 7 = 8 - 7 = 1.

Since the lateral limits are equal as the numeric value of the function, the function is continuous at x = 2.

More can be learned about continuity at https://brainly.com/question/24637240

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