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Question
Identify discontinuities for the following function as either a jump or a removable discontinuity :
f(x) = `(x^2 - 10x + 21)/(x - 7)`
Solution
Given, f(x) = `(x^2 - 10x + 21)/(x - 7)`
It is rational function and is discontinuous if
x – 7 = 0 i.e., x = 7
∴ f(x) is continuous for all x ∈ R, except at x = 7.
∴ f(7) is not defined.
Now, `lim_(x -> 7) "f"(x) = lim_(x -> 7) (x^2 - 10x + 21)/(x - 7)`
= `lim_(x -> 7) ((x - 7)(x - 3))/(x - 7)`
= `lim_(x -> 7) (x - 3) ...[(because x -> 7"," therefore x ≠ 7),(therefore x - 7 ≠ 0)]`
= 7 – 3
= 4
Thus `lim_(x -> ) "f"(x)` exist but f(7) is not defined
∴ f(x) has a removable discontinuity.
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