Question

Examine the following function for continuity:

f(x) = `1/(x - 5)`, x ≠ 5

Answer 1

Let a be a real number, then,

`lim_(x->a^+)` f(x) = `lim_(h->0) 1/(a + h - 5) = 1/(a - 5)`

`lim_(x->a^-)` f(x) = `lim_(h->0) 1/(a - h - 5) = 1/(a-5)`

f(a) = `1/(a-5)`

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

Hence, the given function f(x) = `1/(x - 5)` is continuous at all points except at x = 5.