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: Check left and right limits separately, even though it is a polynomial:
For $x$ slightly less than 3, $f(x)=2x^2-1$ approaches $2(3)^2-1=17$. For $x$ slightly more than 3, it also approaches $17$, since squaring is continuous.

Step 2: Compare both one-sided limits:
LHL $=$ RHL $=17$, so the two-sided limit exists and equals 17.

Step 3: Compare with the function value:
$f(3)=2(9)-1=17$, which matches the limit exactly.

Final Answer:
Since limit $=f(3)=17$, the function is continuous at $x=3$. \[ \boxed{\text{Continuous},\ f(3)=17} \]