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Question
If f(x) = `{{:((x^3 + x^2 - 16x + 20)/(x - 2)^2",", x ≠ 2),("k"",", x = 2):}` is continuous at x = 2, find the value of k.
Solution
Given f(2) = k.
Now, `lim_(x -> 2) "f"(x) = lim_(x -> 2^+) "f"(x)`
= `lim_(x -> 2) (x^3 + x^2 - 16x + 20)/(x - 2)^2`
= `lim_(x -> 2) ((x - 5)(x - 2)^2)/(x - 2)^2`
= `lim_(x -> 2) (x + 5)`
= 7
As f is continuous at x = 2, we have
`lim_(x -> 2) "f"(x)` = f(2)
⇒ k = 7.
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