The value of f0- so that the function fx - a 2- ax - x 2- a 2- ax - x 2 a - x - a - x becomes continuous for all x- given bya a3 - 2b a1 - 2c -a1 - 2d -a3 - 2

(c) a12 Given: fx=a2 ax+x2 a2+ax+x2a+x a x #8658; #160;fx=a2 ax+x2 a2+ax+x2a2 ax+x2+a2+ax+x2a+x a xa2 ax+x2+a2+ax+x2 #8658; #160;fx=a2 ax+x2 a2+ax+x ...

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Standard XII

Mathematics

Continuity in an Interval

The value of ...

Question

The value of f (0), so that the function
$f (x) = \frac{\sqrt{a^{2} - a x + x^{2}} - \sqrt{a^{2} + a x + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}}$ becomes continuous for all x, given by
(a) a³^/2
(b) a¹^/2
(c) −a¹^/2
(d) −a³^/2

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Solution

(c) $- a^{\frac{1}{2}}$

Given: $f (x) = \frac{\sqrt{a^{2} - a x + x^{2}} - \sqrt{a^{2} + a x + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}}$

$\Rightarrow f (x) = \frac{(\sqrt{a^{2} - a x + x^{2}} - \sqrt{a^{2} + a x + x^{2}}) (\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})}{(\sqrt{a + x} - \sqrt{a - x}) (\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})} \Rightarrow f (x) = \frac{(a^{2} - a x + x^{2} - (a^{2} + a x + x^{2}))}{(\sqrt{a + x} - \sqrt{a - x}) (\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})} \Rightarrow f (x) = \frac{(- 2 a x) (\sqrt{a + x} + \sqrt{a - x})}{(\sqrt{a + x} - \sqrt{a - x}) (\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}}) (\sqrt{a + x} + \sqrt{a - x})} \Rightarrow f (x) = \frac{(- 2 a x) (\sqrt{a + x} + \sqrt{a - x})}{(a + x - a + x) (\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})} \Rightarrow f (x) = \frac{(- 2 a x) (\sqrt{a + x} + \sqrt{a - x})}{(2 x) (\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})} \Rightarrow f (x) = \frac{- a (\sqrt{a + x} + \sqrt{a - x})}{(\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})}$

If $f (x)$ is continuous for all x, then it will be continuous at x = 0 as well.

So, if $f (x)$ is continuous at x = 0, then
$\lim_{x \to 0} f (x) = f (0)$

$\Rightarrow \lim_{x \to 0} [\frac{- a (\sqrt{a + x} + \sqrt{a - x})}{(\sqrt{a^{2} - a x + x^{2}} + \sqrt{a^{2} + a x + x^{2}})}] = f (0) \Rightarrow [\frac{- 2 a (\sqrt{a})}{(\sqrt{a^{2}} + \sqrt{a^{2}})}] = f (0) \Rightarrow [\frac{- 2 a (\sqrt{a})}{(a + a)}] = f (0) \Rightarrow f (0) = - \sqrt{a}$

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The value of f (0), so that the function fx=a2-ax+x2-a2+ax+x2a+x-a-x becomes continuous for all x, given by (a) a3/2 (b) a1/2 (c) −a1/2 (d) −a3/2

The value of f (0), so that the function
$f (x) = \frac{\sqrt{a^{2} - a x + x^{2}} - \sqrt{a^{2} + a x + x^{2}}}{\sqrt{a + x} - \sqrt{a - x}}$ becomes continuous for all x, given by
(a) a³^/2
(b) a¹^/2
(c) −a¹^/2
(d) −a³^/2