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Find k if the following function is continuous at the points indicated against them: f(x)=(1+kx)1x , for x ≠ 0 = e32 , for x = 0, at x = 0 - Mathematics and Statistics

Sum

Find k if the following function is continuous at the points indicated against them:

`f(x) = (1 + kx)^(1/x)`  , for x ≠ 0

       = `e^(3/2)`                  ,  for x = 0, at x = 0

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Solution

f is continuous at x = 0

∴ `lim_(x→0) "f"(x)` = f(0)

∴ `lim_(x→0) (1 + "k"x)^(1/x) = "e"^(3/2)`

∴ `lim_(x→0) [(1 + "k"x)^(1/("k'"x)]]^"k" = "e"^(3/2)`

∴ `"e"^"k" = "e"^(3/2) ....[lim_(x→0) (1 + x)^(1/x) = "e"]`

∴ k = `3/2`

Concept: Examples of Continuous Functions Whereever They Are Defined
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 11th Standard Maharashtra State Board
Chapter 8 Continuity
Miscellaneous Exercise 8 | Q II. (3) | Page 113
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