Question

Examine the continuity of the following:

`|x - 2|/|x + 1|`

Answer 1

Let f(x) = `|x - 2|/|x + 1|`

f(x) is defined for all points of R except at x = – 1.

∴ f(x) is defined for all points of R – {– 1}.

Let x₀ be an arbitrary point in R – {– 1}.

Then `lim_(x -> x_0) f(x) =  lim_(x ->x_0) |x - 2|/|x + 1|`

= `|x_0 - 2|/|x_0 + 1|`  .......(1)

`f(x_0) = |x_0 - 2|/|x_0 + 1|`  .......(2)

From equation (1) and (2) we have

`lim_(x -> x_0) f(x) = f(x_0)`

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

Since x = x₀ is an arbitrary point in R – {– 1}, the above result is true for all points in R – {– 1).

∴ f(x) is continuous at all points of R – {– 1}.