Question

What is the continuity of [x] at x = 3.5?
(A) Continuous at the x = 3.5
(B) Discontinuous at the x = 3.5

(C) Function doesn’t exist
(D) None of the above
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Answer 1

Answer:

We can rewrite f(x) as ,

f(x) = x-3 for x≥3

= 3-x for x<3

LHL of f(x) at x=3 is 0.

RHL of f(x) at x=3 is 0.

So, f(x) is continuous at x=3.

RHD of f(x) at x=3 ,

So, f(x) is not differentiable at x=3.

hope it helps you

Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

The continuity of [x] at x = 3.5

(A) Continuous at the x = 3.5

(B) Discontinuous at the x = 3.5

(C) Function doesn’t exist

(D) None of the above

EVALUATION

Let f(x) be the given function

Then f(x) = [ x ]

We have to check the continuity of f(x) at x = 3.5

We have

LHL

$\displaystyle \sf = \lim_{x \to 3.5 - } f(x)$

$\displaystyle \sf = \lim_{x \to 3.5 - } [x]$

$\displaystyle \sf = 3$

RHL

$\displaystyle \sf = \lim_{x \to 3.5 + } f(x)$

$\displaystyle \sf = \lim_{x \to 3.5 + } [x]$

$\displaystyle \sf =3$

Again

$\displaystyle \sf f(3.5) = [3.5] = 3$

Thus we have

$\displaystyle \sf \lim_{x \to 3.5 - } f(x) = \lim_{x \to 3.5 + } f(x) = f(3.5)$

Therefore , f(x) = [ x ] is continuous at 3.5

FINAL ANSWER

Hence the correct option is

(A) Continuous at the x = 3.5

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