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Differential Calculus
f(x) = lt.(n --> infinity) {[x^2] + [(2x)^2] + [(3x)^2] +.........+[(nx)^2]}/nAt what values of x is f(x) continous? [.} represents greatest integer function.a) {-infinity,infinity} ~ {0}b) {-infinity,infinity}c) {-infinity,infinity} ~ Id) {-infinity,infinity} ~ {0,1}
Comments (6)
(x^2)-1
[x^2] 
x^2
(2x)^2 -1
[(2x)^2]
(2x)^2
(3x)^2 -1
[(3x)^2]
(3x)^2
.... .... ....
.... ..... ....
..... ..... ....
(nx)^2 -1
[(nx)^2] 
(nx)^2
adding all these u will get
( x^ +(2x)^2 +(3x)^2+...................+(nx)^2 ) -(1+1+1+1+1.........+1)
g(x) 
( x^2 +(2x)^2+......................+(nx)^2 )
----upto n times-----
x^2( 1^2 +2^2+.................+n^2) - n 
g(x) 
x^2( 1^2+2^2+................+n^2)
x^2 * 
-n 
g(x) 
x^2 * 
x^2* 
-n 
g(x) 
x^2*
DIVIDE BY n ALL SIDES
X^2*
-1 
x^2
x^2 * n^2* 
-1
x^2 * n^2 *
clearly we can see that if x tends to zero than a case of (
) will come which is undetermined hence i think option a is correct continues for all x except 0
let f(x) = lim n tends to infinity g(x)/n
(x^2)-1
[x^2] 
x^2
(2x)^2 -1
[(2x)^2]
(2x)^2
(3x)^2 -1
[(3x)^2]
(3x)^2
.... .... ....
.... ..... ....
..... ..... ....
(nx)^2 -1
[(nx)^2] 
(nx)^2
adding all these u will get
( x^ +(2x)^2 +(3x)^2+...................+(nx)^2 ) -(1+1+1+1+1.........+1)
g(x) 
( x^2 +(2x)^2+......................+(nx)^2 )
----upto n times-----
x^2( 1^2 +2^2+.................+n^2) - n 
g(x) 
x^2( 1^2+2^2+................+n^2) x^2 * 
-n 
g(x) 
x^2 * 
x^2* 
-n 
g(x) 
x^2*
DIVIDE BY n ALL SIDES
X^2*
-1 
x^2
x^2 * n^2* 
-1
x^2 * n^2 *
clearly we can see that if x tends to zero than a case of (
) will come which is undetermined hence i think option a is correct continues for all x except 0
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is b option that is (-infinity ,infinity) correct?