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Limits and Continuity Test - 3
Result
Limits and Continuity Test - 3

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  • Question 1/10
    1 / -0

    Solutions

     

  • Question 2/10
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    The function y = f ( x ) is defined by x = 2t - | t |, y = t2 + t | t |, t ∈ R in the interval x ϵ [ - 1, 1 ] then

    Solutions

     

  • Question 3/10
    1 / -0

    Consider the function f ( x ) = x - | x - x2 |, - 1 ≤ x ≤ 2. The points of discontinuities of f ( x ) for x ∈ [ - 1, 2 ] are

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  • Question 4/10
    1 / -0

    Solutions

     

  • Question 5/10
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    Solutions

     

  • Question 6/10
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    Solutions

     

  • Question 7/10
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    Suppose ' f ' is continuous function from R to R and f ( f ( a ) ) = a for some a ∈ R then the equation f ( x ) = x has

    Solutions

    If f ( a ) = a, then obviously x = a is the solution

    Let f ( a ) > a and g ( x ) = f ( x ) - x 

    then g ( a ) > 0 and

    g ( f ( a ) ) = f ( f ( a ) ) - f ( a ) = a - f ( a ) < 0

    Since g(x) is continuous, so at least for one

    c ∈ ( a, f ( a ) ), g ( c ) = 0.

    Similarly we can argue for f ( a ) < a.

    The correct answer is: at least three solutions

  • Question 8/10
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  • Question 9/10
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    Let f ( x ) be a continuous function ∀ x ∈ R, f ( 0 ) = 1 and f ( x ) ≠ x for any x ∈ R then

    Solutions

    Let g ( x ) = f ( x ) - x, so g ( x ) is continuous and g ( 0 ) = 1.

    Now it is given that g ( x ) ≠ 0 for any x ∈ R so,

    g ( x ) > 0 ∀ x ∈ R 

    i.e., f ( x ) > x ∀ x ∈ R

    ⇒ f ( f ( x ) ) > f ( x ) > x ∀ x ∈ R.

    The correct answer is: f ( f ( x ) ) > x ∀ x ∈ R

  • Question 10/10
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    Solutions

     

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