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Question
Find k if following function is continuous at the point indicated against them:
f(x) `{:(= ((5x - 8)/(8 - 3x))^(3/(2x - 4))",", "for" x ≠ 2),(= "k"",", "for" x = 2):}}` at x = 2
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Solution
f(x) is continuous at x = 2
∴ f(2) = `lim_(x -> 2) "f"(x)`
∴ k = `lim_(x -> 2) ((5x - 8)/(8 - 3x))^(3/(2x - 4))`
Put x – 2 = h
∴ x = 2 + h
As x → 2, h → 0
∴ k = `lim_("h" -> 0) [(5(2 + "h") - 8)/(8 - 3(2 + "h"))]^(3/(2"h"))`
= `lim_("h" -> 0) ((10 + 5"h" - 8)/(8 - 6 - 3"h"))^(3/(2"h"))`
= `lim_("h" -> 0) ((2 + 5"h")/(2 - 3"h"))^(3/(2"h"))`
= `lim_("h" -> 0) [(2(1 + (5"h")/2))/(2(1 - (3"h")/2))]^(3/(2"h"))`
= `lim_("h" -> 0) (1 + (5"h")/2)^(3/(2"h"))/((1 - (3"h")/2)^(3/(2"h"))`
= `(lim_("h" -> 0) [(1 + (5"h")/2)^(2/(5"h"))]^(5/2 xx 3/2))/(lim_("h" -> 0)[(1 - (3"h")/2)^((-2)/(3"h"))]^((-3)/2 xx 3/2)`
= `"e"^(15/4)/"e"^((-9)/4) ...[(because "h" -> 0"," (5"h")/2 -> 0"," (-3"h")/2 -> 0),("and" lim_(x -> 0) (1 + x)^(1/x) = "e")]`
= `"e"^(24/4)`
= e6
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