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Question
Examine the continuity of the following:
`(x^2 - 16)/(x + 4)`
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Solution
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}.
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