f x - left matrix- x - a - sin frac 1 x - a - - if - x lneq 0 10- - if - x - aCheck if f x is continuous at x - a-

We have, f(x)={ |x a| sin 1/x a, amp; if x≠ 0 0, amp; if x=a .at x=a At x=a, LHL = x→ a^ lim|x a| sin 1/x a =h→ 0lim |a h a|sin ( 1/a h a ) =h→ 0lim ...

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Byju's Answer

Standard XII

Mathematics

Parametric Differentiation

f x = left ma...

Question

$f (x) = {\begin{matrix} | x - a | s i n \frac{1}{x - a}, & i f x \neq 0 0, & i f x = a \end{matrix} a t x = a$

Check if f(x) is continuous at x=a.

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Solution

We have, $f (x) = {\begin{matrix} | x - a | s i n \frac{1}{x - a}, & i f x \neq 0 0, & i f x = a \end{matrix} a t x = a$
At x=a,
LHL $= l i m x \to a^{-} | x - a | s i n \frac{1}{x - a} = l i m h \to 0 | a - h - a | s i n (\frac{1}{a - h - a}) = l i m h \to 0 - h s i n (\frac{1}{h})$
$= 0 \times$ [an oscillating number between -1 and 1]=0
RHL $= l i m x \to a^{+} | x - a | s i n (\frac{1}{x - a}) = l i m h \to 0 | a + h - a | s i n (\frac{1}{a + h - a}) = l i m h \to 0 h s i n \frac{1}{h}$
$= 0 \times$ [an oscillating number between -1 and 1]=0
and f(a)=0
$∴$ LHL = RHL f(a)
So, f(x) is continuous at x=a

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f(x)={|x−a|sin1x−a,if x≠00,if x=aat x=a Check if f(x) is continuous at x=a.

$f (x) = {\begin{matrix} | x - a | s i n \frac{1}{x - a}, & i f x \neq 0 0, & i f x = a \end{matrix} a t x = a$

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