If fx- eft begin matrix Veft - frac x- 2 a right -a- - when - x-a0 - -when - x-alend matrix right - then

The correct option is A f(x) is continuous at x = af(a) = 0 lim_x → a f(x)=lim_x → a ( x^2/a a )=lim_k → 0{(a h)^2/a a }=0 and lim_x → a+f(x)=lim_x → 0{ a (a ...

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Byju's Answer

Standard XII

Mathematics

Differentiability in an Interval

If fx=∪ eft b...

Question

If $f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (\frac{x^{2}}{a}) - a, & w h e n & x < a 0, & w h e n & x = a a - (\frac{x^{2}}{a}), & w h e n & x > a \end{matrix}$ , then

l i m_{x \to a} f (x) = a

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f(x) is continuous at x = a

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f(x) is discontinuous at x = a

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None of these

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Solution

The correct option is B f(x) is continuous at x = a
f(a) = 0
$l i m_{x \to a -} f (x) = l i m_{x \to a -} (\frac{x^{2}}{a} - a) = {lim}_{k \to 0} {\frac{(a - h)^{2}}{a} - a} = 0$
and $l i m_{x \to a +} f (x) = l i m_{x \to 0} {a - \frac{(a + h)^{2}}{a}} = 0$
Hence it is continuous at x = a.

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If f(x)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(x2a)−a,whenx<a0,whenx=aa−(x2a),whenx>a, then

The correct option is B f(x) is continuous at x = af(a) = 0 limx→a−f(x)=limx→a−(x2a−a)=limk→0{(a−h)2a−a}=0 and limx→a+f(x)=limx→0{a−(a+h)2a}=0 Hence it is continuous at x = a.

If $f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (\frac{x^{2}}{a}) - a, & w h e n & x < a 0, & w h e n & x = a a - (\frac{x^{2}}{a}), & w h e n & x > a \end{matrix}$ , then

The correct option is B f(x) is continuous at x = a
f(a) = 0
$l i m_{x \to a -} f (x) = l i m_{x \to a -} (\frac{x^{2}}{a} - a) = {lim}_{k \to 0} {\frac{(a - h)^{2}}{a} - a} = 0$
and $l i m_{x \to a +} f (x) = l i m_{x \to 0} {a - \frac{(a + h)^{2}}{a}} = 0$
Hence it is continuous at x = a.