Question

Determine whether the following functions are continuous at a given point. Show your complete solution.

1. ƒ(𝑥)=3𝑥2−4𝑥+2at𝑥=2

2. ƒ(𝑥)=𝑥2−6𝑥−3at𝑥=4

Answer 1

For a function f to be continuous, at a given point, then:

• The function must be defined at a point a to be continuous at that point x = a.
• The limit of the function f(x) should exist at the point x = a,
• The value of the function f(x) at that point, i.e. f(a) must equal the value of the limit of f(x) at x = a.

1. $ƒ(𝑥)=3𝑥^2−4𝑥+2 \text{ at }𝑥=2$

(i) Substituting x=2 in f(x):

$f(2) = 3(2)^2 -4(2)+2 =3(4)-4(2)+2 = 6$

(ii) Limit at x=2

$$\begin{aligned} \lim_{x \rightarrow2} ƒ(𝑥) &= \lim_{x \rightarrow2} (3𝑥^2−4𝑥+2)\\ & = 3(2)^2 -4(2)+2\\ &=3(4)-4(2)+2\\ &= 6\\ &=f(2) \end{aligned}$$

$\therefore ƒ(𝑥)=3𝑥^2−4𝑥+2 \text{ is continuous at } 𝑥=2$

2. $ƒ(𝑥)=𝑥^2−6𝑥−3 \text{ at } 𝑥=4$

(i) substituting x=4 in f(x)

$ƒ(4)=(4)^2−6(4)−3 = 16-24-3 = -11$

(ii) Limit at x =4

$$\begin{aligned} \lim_{x \rightarrow 4} f(x) &= \lim_{x \rightarrow 4} (𝑥^2−6𝑥−3)\\ &= (4)^2−6(4)−3\\ & = 16-24-3\\ & = -11\\ & = f(4) \end{aligned}$$

$\therefore ƒ(𝑥)=𝑥^2−6𝑥−3 \text{ is continuous at } 𝑥=4$