Question

Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

3. at x = 5

x² - 3 if x ≥ 5

4x + 2 if x < 5

Answer 1

$f(x)=\{\begin{matrix} x^2-3 & \text{if }x\geq 5, \\ 4x+2 & \text{if }x< 5. \end{matrix}$

The function is a continuous at a given point x=a when f(a) exists and $lim_{x \to a}f(x)=f(a).$

$f(5)=5^2-3=25-3=22.$

$lim_{x\to 5^-}f(x)=lim_{x\to 5^-}(4x+2)=4\cdot 5+2=22,$

$lim_{x\to 5^+}f(x)=lim_{x\to 5^-}(x^2-3)=25-3=22,$

so

$lim_{x\to 5^-}f(x)=lim_{x\to 5^+}f(x)=22=f(5),$

and the function is continuous at x=5.

Answer: the function is continuous at x=5.