Question

Examine the following function for continuity:

f(x) = `(x^2 - 25)/(x + 5)`, x ≠ −5

Answer 1

Let f(x) = `(x^2 - 25)/(x + 5)`

= `((x + 5) (x - 5))/(x + 5)`

= x − 5

∵ f(x) = x − 5

Let ‘a’ be a real number, then,

`lim_(x->a^+)` f(x) = `lim_(h->0)` (a + h) − 5 = a − 5

`lim_(x->a^-)` f(x) = `lim_(h->0)` (a − h) − 5 = a − 5

Also, f(a) = a − 5

∵ `lim_(x->a^+)` f(x) = `lim_(x->a^-)` f(x) = f(a)

Hence, the given function f(x) = x − 5 is continuous at every point of its domain.