Question

Check the continuity of f(x) = x²-4/x-2 at x=2 where f(2)=4.​

Answer 1

Answer:

f(x) is continuous at x=2.

Step-by-step explanation:

at x= 2+,

x=2+h, h tends 0.

lim h=0 f(x) = (2+h+2)(2+h-2)/(2+h-2)

=4.

lly, LHC = 4.

f(2) = 4.

Hence,f(x) is continuous function.

Answer 2

EXPLANATION.

$\sf \implies \displaystyle f(x) = \frac{x^{2} - 4}{x - 2} \ \ \ at \ \ \ x = 2$

As we know that,

A function f(x) is said to be continuous at x = a if,

L.H.L = R.H.L = f(a) = A finite quantity.

f(a⁻) = f(a⁺) = f(a).

Using this concept in the equation, we get.

$\sf \implies \displaystyle \lim_{x \to 2^{-} } \bigg(\frac{x^{2} - 4}{x - 2}\bigg)$

$\sf \implies \displaystyle \lim_{x \to 2^{-} } \frac{(x - 2)(x + 2)}{(x - 2).}$

$\sf \implies \displaystyle \lim_{x \to 2^{-} } (x + 2)$

Put the value of x = 2 in the equation, we get.

$\sf \implies \displaystyle 2 + 2 = 4$

$\sf \implies \displaystyle \lim_{x \to 2^{+} } \bigg(\frac{x^{2} - 4}{x - 2}\bigg)$

$\sf \implies \displaystyle \lim_{x \to 2^{+} } \frac{(x - 2)(x + 2)}{(x - 2).}$

$\sf \implies \displaystyle \lim_{x \to 2^{+} } (x + 2)$

Put the value of x = 2 in the equation, we get.

$\sf \implies \displaystyle 2 + 2 = 4$

As we can see that,

⇒ f(a⁻) = f(a⁺) = f(a).

⇒ f(2⁻) = f(2⁺) = f(2).

Also, f(2) = 4.

The function is continuous at f(2) = 4.

                                                                                                                 

MORE INFORMATION.

Continuity in an interval.

(1) f(x) is said to continuous in [a, b].

(a) ∨ α ∈ (a, b).

f(α⁻) = f(α⁺) = f(α) = A finite quantity.

Continuous in all interior points.

(b) f(α⁺) = f(α) = A finite quantity.

f(x) is continuous at x = a.

Right continuous.

(c) f(b⁻) = f(b) = A finite quantity.

f(x) is continuous at x = b.

Left continuous.

(2) f(x) is continuous in [a, b).

∨ α ∈ [a, b).

(a) f(α⁻) = f(α) = f(α) = A finite quantity.

(b) f(a⁺) = f(a) = A finite quantity.

(3) f(x) is continuous in (a, b].

∨ α ∈ (a, b].

(a) f(α⁻) = f(α⁺) = f(α) = A finite quantity.

(b) f(b⁻) = f(b) = A finite quantity.

(4) f(x) is continuous in (a, b).

∨ α ∈ [a, b].

f(α⁻) = f(α⁺) = f(α) = A finite quantity.