Question

how are \\(\lim_{x \to a^-} p(x)\\) and \\(\lim_{x \to a^+} p(x)\\) calculated if \\(p\\) is a polynomial function?

choose the correct answer below.

a. \\(\lim_{x \to a^-} p(x) = \lim_{x \to a^+} p(x) = p(a)\\)
b. \\(\lim_{x \to a^-} p(x) = \lim_{x \to a^+} p(x) = 0\\)
c. \\(\lim_{x \to a^-} p(x) = \lim_{x \to a^+} p(x) = a\\)
d. either of the limits may not exist.

Explanation:

Identify the properties of polynomial functions

Polynomial functions are continuous everywhere on the real line \(\mathbb{R}\).

Apply the definition of continuity

For any polynomial function \(p(x)\) and any real number \(a\):

$$ \lim_{x \to a} p(x) = p(a) $$

Relate to one-sided limits

Since the two-sided limit exists and equals \(p(a)\), both the left-hand and right-hand limits must exist and equal this same value:

$$ \lim_{x \to a^-} p(x) = \lim_{x \to a^+} p(x) = p(a) $$

Answer:

• (A) \(\lim_{x \to a^-} p(x) = \lim_{x \to a^+} p(x) = p(a)\) (Correct answer)
• (B) \(\lim_{x \to a^-} p(x) = \lim_{x \to a^+} p(x) = 0\)
• (C) \(\lim_{x \to a^-} p(x) = \lim_{x \to a^+} p(x) = a\)
• (D) Either of the limits may not exist.