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प्रश्न
Examine the continuity of the following:
`|x - 2|/|x + 1|`
उत्तर
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}.
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