Determine if f defined by f(x)= ⎩⎨⎧x2sinx1, if x=00, if x=0 is a continuous function ?

Question

Determine if f defined by f(x)= ⎩⎨⎧​x2sinx1​, if x=00, if x=0​ is a continuous function ?

Filo · Accepted Answer

f(x)=⎩⎨⎧​x2sinx1​;x=00;x=0  ​At x=0A function is continuous at x=0if L.H.L=R.H.L=f(0)limx→0−​f(x)=limx→0+​f(x)=f(0)L.H.L=limx→0−​f(x)=limx→0−​x2sinx1​Put x=0−hAs x→0,0−h→0,h→0=limh→0​(0−h)2sin(0−h)1​=limh→0​(−h)2sin(−h)1​=limh→0​h2sin(−h)1​=−limh→0​h2sinh1​ as sin(−x)=−sinxWe know that −1≤sinθ≤1⇒−1≤sinh1​≤1∴sinh1​ is a finite valueLet sinh1​=k=−limh→0​h2kPutting h=0=0×k=0∴L.H.L=0R.H.L=limx→0+​f(x)=limx→0+​x2sinx1​Put x=0+hAs x→0,0+h→0,h→0=limh→0​(0+h)2sin(0+h)1​=limh→0​(h)2sin(h)1​=limh→0​h2sin(h)1​=limh→0​h2sinh1​ We know that −1≤sinθ≤1⇒−1≤sinh1​≤1∴sinh1​ is a finite valueLet sinh1​=k=limh→0​h2kPutting h=0=0×k=0∴R.H.L=0and f(0)=0Thus,L.H.L=R.H.L=f(0)Hence f(x) is continuous for all real value of x

Question Text	Determine if $f$ defined by $f (x) =$ $⎩ ⎨ ⎧ x^{2} sin \frac{1}{x}, if x \neq = 0 0, if x = 0$ is a continuous function ?
Answer Type	Text solution:1
Upvotes	150

Determine if $f$ defined by
$f (x) =$ $⎩ ⎨ ⎧ x^{2} sin \frac{1}{x}, if x \neq = 0 0, if x = 0$
is a continuous function ?

Text solutionVerified

Practice questions from similar books

Determine if f defined by f(x)= ⎩⎨⎧​x2sinx1​, if x=00, if x=0​ is a continuous function ?

Text solutionVerified

Practice questions from similar books

Determine if $f$ defined by
$f (x) =$ $⎩ ⎨ ⎧ x^{2} sin \frac{1}{x}, if x \neq = 0 0, if x = 0$
is a continuous function ?