Question

Determine if the following function is continuous at 𝑥 = 2. 𝑓(𝑥) = { 2𝑥 −3, 𝑥 ≤ 2 𝑥2, 𝑥 > 2?

Answer 1

Kindly refer to the Discussion in Explanation.

Explanation:

For `f` to be continuous at `x=2`, we must have,

`lim_(x to 2-) f(x)=f(2)=lim_(x to 2+)......(gamma).`

Now, as `x to 2-, x lt 2,"so that, "f(x)=2x-3...[because,"defn.]"`

`:. lim_(xto 2-) f(x)=lim_(x to 2-)(2x-3)=2(2)-3=1......(gamma_l)`.

On the other hand,

as `x to 2+, x gt 2. :." by defn., "f(x)=x^2`.

`:. lim_(xto2+)f(x)=lim_(x to 2+)x^2=2^2=4......(gamma_r)`.

Thus, `(gamma), (gamma_l) and (gamma)_r`, altogether

`rArr lim_(x to 2-)f(x)=1!=4=lim_(x to 2+)f(x)`.

Evidently, `f` is not continuous at `x=2`.