Question

Examine the continuity of the following:

`(x^2 - 16)/(x + 4)`

Answer 1

Let fx) = `(x^2 - 16)/(x + 4)`

f(x) is not defined at x = – 4

∴ f(x) is defined for all points of R – {– 4}.

Let x₀ be an arbitrary point in R – {– 4}.

Then `lim_(x -> x_0) f(x) =  lim_(x -> x_0) (x^2 - 16)/(x + 4)`

= `(x_0^2 - 16)/(x_0 + 4)` ........(1)

`f(x_0) = (x_0^2 - 16)/(x_0 + 4)`  ........(2)

From equation (1) and (2) we have

`lim_(x -> x_0) (x^2 - 16)/(x + 4)= f(x_0)`

Thus the limit of the function f(x) exist at x = x₀ and is equal to the value of the function f(x) at x = x₀.

Since x₀ is an arbitrary point in R – {– 4} the above result is true for all points in R – {– 4}.

∴ f(x) is continuous at all points of R – {– 4}.