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Question
If f(x) = `{{:("m"x + 1",", "if" x ≤ pi/2),(sin x + "n"",", "If" x > pi/2):}`, is continuous at x = `pi/2`, then ______.
Options
m = 1, n = 0
m = `("n"pi)/2 + 1`
n = `("m"pi)/2`
m = n = `pi/2`
Solution
If f(x) = `{{:("m"x + 1",", "if" x ≤ pi/2),(sin x + "n"",", "If" x > pi/2):}`, is continuous at x = `pi/2`, then n = `("m"pi)/2`.
Explanation:
Given that: f(x) = `{{:("m"x + 1",", "if" x ≤ pi/2),(sin x + "n"",", "If" x > pi/2):}`, is continuous at x = `pi/2`
L.H.L. = `lim_(x -> pi^-/2) ("m"x + 1)`
= `lim_("h" -> 0) ["m"(pi/2 - "h") + 1]`
= `("m"pi)/2 + 1`
R.H.L. = `lim_(x -> pi^+/2) (sinx + "n")`
= `lim_("h" -> 0) [sin(pi/2 + "h") + pi]`
= `lim_("h" -> 0) cos "h" + "n"`
= 1 + n
When f(x) is continuous at x = `pi/2`
∴ L.H.L. = R.H.L.
`("m"pi)/2 + 1` = 1 + n
⇒ n = `("m"pi)/2`.
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