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Question
Find the values of k so that the function f is continuous at the indicated point.
`f(x) = {(kx +1, if x<= pi),(cos x, if x > pi):} " at x " = pi`
Solution
`f(x) = {(kx +1, if x<= pi),(cos x, if x > pi):} " at x " = pi`
If f(x) is continuous at `x = pi` it implies:
`f(pi) = lim_(x -> pi^+) f(x) = lim_(x -> pi^-) f(x)`
`=> k (pi) + 1 = cos (pi) = k (pi) + 1`
`=> k (pi) + 1 = cos (pi)`
`=> k (pi) + 1 = - 1`
`=> k = (-2)/pi`
That is, for the quantity `k = (-2)/pi` this function is continuous at `x = pi`.
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