Home
Class 12
MATHS
If f (x) = [{:((ae^(sin x )+be ^(-sinx)-...

If `f (x) = [{:((ae^(sin x )+be ^(-sinx)-c)/(x ^(2)),,, x ne0),(2, ,, x =0):}` is continous at `x=0,` then:

A

`a =b=c`

B

`a=2b=3c`

C

`a=b=2c`

D

`2a=2b=c`

Text Solution

AI Generated Solution

The correct Answer is:
To determine the conditions under which the function \[ f(x) = \begin{cases} \frac{ae^{\sin x} + be^{-\sin x} - c}{x^2} & \text{if } x \neq 0 \\ 2 & \text{if } x = 0 \end{cases} \] is continuous at \(x = 0\), we need to ensure that: \[ \lim_{x \to 0} f(x) = f(0) = 2 \] ### Step 1: Calculate the limit as \(x\) approaches 0 For \(x \neq 0\), we consider the limit: \[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{ae^{\sin x} + be^{-\sin x} - c}{x^2} \] ### Step 2: Evaluate the numerator at \(x = 0\) As \(x\) approaches 0, we can use the fact that \(\sin(0) = 0\): \[ ae^{\sin(0)} + be^{-\sin(0)} - c = a \cdot 1 + b \cdot 1 - c = a + b - c \] ### Step 3: Apply L'Hôpital's Rule Since both the numerator and denominator approach 0 as \(x\) approaches 0, we can apply L'Hôpital's Rule: \[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{d}{dx}(ae^{\sin x} + be^{-\sin x} - c) \bigg/ \frac{d}{dx}(x^2) \] ### Step 4: Differentiate the numerator and denominator The derivative of the numerator: \[ \frac{d}{dx}(ae^{\sin x} + be^{-\sin x}) = a e^{\sin x} \cos x - b e^{-\sin x} \cos x \] The derivative of the denominator: \[ \frac{d}{dx}(x^2) = 2x \] ### Step 5: Evaluate the limit again Now we evaluate: \[ \lim_{x \to 0} \frac{a e^{\sin x} \cos x - b e^{-\sin x} \cos x}{2x} \] Substituting \(x = 0\): \[ = \frac{a \cdot 1 \cdot 1 - b \cdot 1 \cdot 1}{0} = \frac{a - b}{0} \] ### Step 6: Set the numerator equal to zero For the limit to exist, the numerator must also approach 0 as \(x\) approaches 0: \[ a - b = 0 \implies a = b \] ### Step 7: Substitute back into the limit condition Substituting \(a = b\) into the first limit condition: \[ a + b - c = 0 \implies 2a - c = 0 \implies c = 2a \] ### Final Conclusion Thus, we have: 1. \(a = b\) 2. \(c = 2a\) ### Summary of Conditions The conditions for continuity at \(x = 0\) are: - \(a + b = c\) - \(a = b\) - \(c = 2a\)
Promotional Banner

Topper's Solved these Questions

  • CONTINUITY, DIFFERENTIABILITY AND DIFFERENTIATION

    VIKAS GUPTA (BLACK BOOK) ENGLISH|Exercise EXERCISE (ONE OR MORE THAN ONE ANSWER IS/ARE CORRECT)|36 Videos
  • CONTINUITY, DIFFERENTIABILITY AND DIFFERENTIATION

    VIKAS GUPTA (BLACK BOOK) ENGLISH|Exercise EXERCISE (COMPREHENSION TYPE PROBLEMS)|30 Videos
  • COMPOUND ANGLES

    VIKAS GUPTA (BLACK BOOK) ENGLISH|Exercise Exercise-5 : Subjective Type Problems|31 Videos
  • DETERMINANTS

    VIKAS GUPTA (BLACK BOOK) ENGLISH|Exercise EXERCISE-4 : SUBJECTIVE TYPE PROBLEMS|12 Videos

Similar Questions

Explore conceptually related problems

If f(x)={{:(,x^(m)sin((1)/(x)),x ne 0),(,0,x=0):} is a continous at x=0, then

If f (x)= {{:((x-e ^(x)+1-(1-cos 2x))/(x ^(2)), x ne 0),(k,x=0):} is continous at x=0 then which of the following statement is false ?

If f(x)={{:(,((4^(x)-1)^(3))/(sin(x//4)log(1+x^(2)//3)),x ne 0),(,k,x=0):} is a continous at x=0, then k=

If f(x) = (x-e^(x) + cos 2x)/(x^(2)), x ne 0 is continuous at x = 0, then

If f(x)={{:((sinpix)/(5x)",",x ne0),(k,","0):} is continuous at x=0 , then k is equal to

If f (x) {{:((1 - cos 8 x )/(x^(2)) "," x ge 0 ),(lambda ", " x lt 0 ) :} is continous at x = 0 than lambda = ?

f(x)={{:((1-cos2x)/(x^(2)),if x ne 0),(5, if x = 0):} at x = 0 .

The value of k for which f(x)={{:(,[1+x(e^(-1//x^(2)))sin(1/(x^(4)))]^(e^(1//x^(2))),x ne 0),(,k,x=0):} is continuous at x=0, is

Show that the function f(x) ={:{((sin 3x)/(x)", "x ne 0),(1", " x= 0):}, is discontinuous at x=0.

Show that function f(x) given by f(x)={(x sin(1/x),,,x ne 0),(0,,,x=0):} is continuous at x=0

VIKAS GUPTA (BLACK BOOK) ENGLISH-CONTINUITY, DIFFERENTIABILITY AND DIFFERENTIATION-EXERCISE (SUBJECTIVE TYPE PROBLEMS)
  1. If f (x) = [{:((ae^(sin x )+be ^(-sinx)-c)/(x ^(2)),,, x ne0),(2, ,, x...

    Text Solution

    |

  2. Let f (x)= {{:(ax (x-1)+b,,, x lt 1),( x+2,,, 1 le x le 3),(px ^(2) +q...

    Text Solution

    |

  3. If y= sin (8 sin ^(-1) x ) then (1-x ^(2)) (d^(2)y)/(dx ^(2))-x (dy)/...

    Text Solution

    |

  4. If y ^(2) =4ax, then (d^(2) y)/(dx ^(2))=(ka ^(2))/( y ^(3)), where k ...

    Text Solution

    |

  5. The number of values of x , x ∈ [-2,3] where f (x) =[x ^(2)] sin (pix)...

    Text Solution

    |

  6. If f (x) is continous and differentiable in [-3,9] and f'(x) in [-2,8]...

    Text Solution

    |

  7. In f (x)= [{:(cos x ^(2),, x lt 0), ( sin x ^(3) -|x ^(3)-1|,, x ge 0)...

    Text Solution

    |

  8. Consider f(x) =x^(2)+ax+3 and g(x) =x+band F(x) = lim( n to oo) (f...

    Text Solution

    |

  9. Let f (x)= {{:(2-x"," , -3 le x le 0),( x-2"," , 0 lt x lt 4):} Then f...

    Text Solution

    |

  10. If f (x) +2 f (1-x) =x ^(2) +2 AA x in R and f (x) is a differentiable...

    Text Solution

    |

  11. Let f (x)= signum (x) and g (x) =x (x ^(2) -10x+21), then the number o...

    Text Solution

    |

  12. If (d^(2))/(d x ^(2))((sin ^(4)x+ sin ^(2)x+1)/(sin ^(2)x + si n x+1))...

    Text Solution

    |

  13. f (x) =a cos (pix)+b, f'((1)/(2))=pi and int (1//2)^(3//2) f (x) dx =2...

    Text Solution

    |

  14. Let alpha (x) = f(x) -f (2x) and beta (x) =f (x) -f (4x) and alpha '(1...

    Text Solution

    |

  15. Let f (x) =-4.e ^((1-x)/(2))+ (x ^(3))/(3 ) + (x ^(2))/(2)+ x+1 and g ...

    Text Solution

    |

  16. If y=e^(2 sin ^(-1)x) then |((x ^(2) -1) y ^('') +xy')/(y)| is equal t...

    Text Solution

    |

  17. Let f be continuous function on [0,oo) such that lim (x to oo) (f(x)+ ...

    Text Solution

    |

  18. Let f (x)=x+ (x ^(2))/(2 )+ (x ^(3))/(3 )+ (x ^(4))/(4 ) +(x ^(5))/(5)...

    Text Solution

    |

  19. In f (x)= [{:(cos x ^(2),, x lt 0), ( sin x ^(3) -|x ^(3)-1|,, x ge 0)...

    Text Solution

    |

  20. Let f :R to R be a differentiable function satisfying: f (xy) =(f(x)...

    Text Solution

    |

  21. For the curve sinx+siny=1 lying in first quadrant. If underset(xrarr0...

    Text Solution

    |