Question

Examine the continuity of the function \(f(x)=2x^2-1\) at \(x=3\).

Hint:

Check if lim(x→3) f(x) equals f(3); polynomials are continuous everywhere.

Answer 1

Step 1: Key Formula or Approach:
A function \(f\) is continuous at \(x=a\) if \(\displaystyle\lim_{x\to a}f(x) = f(a)\), i.e. the left-hand limit, right-hand limit and the actual function value all agree.

Step 2: Finding \(f(3)\):
\[ f(3) = 2(3)^2-1 = 18-1 = 17 \]

Step 3: Finding the limit as \(x\to3\):
Since \(f(x)=2x^2-1\) is a polynomial, it is continuous everywhere, so \(\displaystyle\lim_{x\to3}f(x) = 2(3)^2-1 = 17\) directly by substitution.

Step 4: Comparing:
The limit (17) equals \(f(3)\) (17).

Final Answer:
\(f(x)\) is continuous at \(x=3\), with \(f(3)=17\). \[ \boxed{\text{Continuous at } x=3,\ f(3)=17} \]