Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Solve it, - For each, let [x] be the greatest integer less than or equal to x. then is equal to : - Limit , continuity and differentiability - JEE Main

 

For each x\epsilon R, let [x] be the greatest integer less than or equal to x. then 

\lim_{x \to0^{-} }\frac{x\left ( \left [ x \right ]+\left | x \right | \right )\sin\left [ x \right ]}{\left | x \right |}

is equal to :

  • Option 1)

     

    1

  • Option 2)

     

    0

  • Option 3)

     

    -sin1

  • Option 4)

     

    sin1

Answers (1)

best_answer

 

Limit of product / quotient -

Limit of product/quotient is the product/quotient of individual limits such that

\lim_{x\rightarrow a}{\left (f(x).g(x) \right )}

=\lim_{x\rightarrow a}{f(x).\lim_{x\rightarrow a}g(x), given that f(x) and g(x) are non-zero finite values

\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}, given that f(x) and g(x) are non-zero finite values

Also\:\lim_{x\rightarrow a}{kf(x)} 

=k\lim_{x\rightarrow a}{f(x)}

 

-

 

 

 

Left hand Limit -

The left hand limit of f(x) as  'x'  tends to 'a' exists and  is equal to l2, if as 'x'  approaches 'a'  through values less than 'a'.


\lim_{x\rightarrow a^{-}}f(x)= l_{2}

- wherein

Where  a-   means ( a - h ) & h\rightarrow 0. Therefore, f(a-h).

 

 

Given that 

\lim_{x\rightarrow 0^-} \frac{x([x] + |x|)\sin[x]}{|x|}

x\rightarrow 0^-

from the concept

[x] = 1 \Rightarrow \lim_{x\rightarrow0^{-}} \frac{x(-x-1)\sin(-1)}{-x} = -\sin(1) \;\;\; (\because |x| = -x)

 


Option 1)

 

1

Option 2)

 

0

Option 3)

 

-sin1

Option 4)

 

sin1

Posted by

admin

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2024 Topper: Farmer’s son Gajare Nilkrishna says ‘keep questioning until you understand’
anam-khan

NIT Puducherry Cut-off 2024: Computer Science and Engineering had highest closing rank last year
anam-khan

What after JEE Main results 2024? All you need to know about JEE Adv, CSAB, JoSAA counselling

Latest Question