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Continuity of a Function

Find the valu...

Question

Find the value of k, so that the function f(x) is continuous at the indicated point
$f (x) = \frac{\sqrt{3} - tan x}{π - 3 x} = k$ at
$x = \frac{π}{3}$

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Solution

For a function to be continuous at $x = \frac{π}{3}$ , the limit must be finite at this point.

But at $x = \frac{π}{3}$ function become $\frac{0}{0}$ form. So using L'Hopital's rule:

$k = lim x \to (π / 3) = \frac{\sqrt{3} - tan x}{π - 3 x}$ = $lim x \to (π / 3) \frac{0 - {sec}^{2} x}{0 - 3} = \frac{- 2^{2}}{- 3} = \frac{4}{3}$

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