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Which one of the following functions is continuous at $x = 3?$

1. $f(x) = \begin{cases} 2,&\text{if$x = 3$} \\ x-1& \text{if$x > 3$}\\ \frac{x+3}{3}&\text{if$x < 3$} \end{cases}$
2. $f(x) = \begin{cases} 4,&\text{if$x = 3$} \\ 8-x& \text{if$x \neq 3$} \end{cases}$
3. $f(x) = \begin{cases} x+3,&\text{if$x \leq 3$} \\ x-4& \text{if$x > 3$} \end{cases}$
4. $f(x) = \begin{cases} \frac{1}{x^3-27}&\text{if$x \neq 3$} \end{cases}$
asked in Calculus | 1.4k views
+1
is it's ans be a?

For continuity, Left hand limit must be equal to right hand limit. For continuity at $x = 3$,

the value of $f(x)$ just above and just below $3$ must be the same.

A. $f(3) = 2. f(3+) = x - 1 = 2. f(3-) =\frac{(x+3)}{3} =\frac{6}{3} = 2.\text{ Hence continuous.}$

B. $f(3) = 4. f(3+) = f(3-) = 8 - 3 = 5. \text{ So, not continuous.}$

C. $f(3) = f(3-) = x + 3 = 6. f(3+) = x - 4 = -1.\text{ So, not continuous.}$

D. $f(3)\text{ is not existing. So, not continuous.}$
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@Arjun Sir, why D is not existing? plzz explain ...am little bit poor in Maths so plzzz
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at x=3 D is approaching infinity and hence function is not defined at x=3. For continuity LHL=RHL=value of the function. Hence not continuous.
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In option B, Limit exists since $\lim_{x \to 3^{-}}{f(x)} = \lim_{x \to 3^{+}}{f(x)}$, but $\lim_{x \to 3}{f(x)}\neq f(3)$
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yes option D is a discontinuous  graph

So this f(x) is continuous at x=3

+1 vote
option A is true because LHL = RHL

LHL=$\lim_{x->3}(x+x)/3=(3+3)/3=2$

RHL=$\lim_{x->3}(x-1)=(3-1)=2$

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