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Question
If the function f (x) = `(15^x - 3^x - 5^x + 1)/(x tanx)`, x ≠ 0 is continuous at x = 0 , then find f(0).
Solution
Function f is continuous at x = 0
∴ `f(0) = lim_(x → 0) f(x)`
= `lim_(x → 0) (15^x - 3^x - 5^x + 1)/(x tanx)`
= `lim_(x → 0) (3^x.5^x - 3^x - 5^x + 1)/(x tanx)`
= `lim_(x → 0) (3^x(5^x - 1) - 1(5^x - 1))/(x tanx)`
= `lim_(x → 0) ((5^x - 1)(3^x - 1))/(x tanx)`
= `(lim_(x → 0) ((5^x - 1)(3^x - 1))/x^2)/(lim_(x → 0) (xtanx)/x^2`
= `((lim_(x → 0) (5^x - 1)/x) (lim_(x → 0) (3^x - 1)/x))/((lim_(x → 0) tanx/x))`
= `((log 5)(log 3))/1`
f(0) = (log5)(log3)
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