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Evaluate lim(x to 0) (1 -cos x)/(x^(2))...

Evaluate `lim_(x to 0) (1 -cos x)/(x^(2))`

Text Solution

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We have, `underset(x to 0)(to) (1 - cos x)/(x) = underset(x to 0)(lim) (2 sin^(2) x//2)/(x^(2)) = 2 underset(x to 0)(lim) (sin x//2 .sin x//2)/(x)`
`= 2 underset(x to 0)(lim) (sin (x)/(2))/(2.(x)/(2)) underset(x to 0)(lim) (sin (x)/2)/(2.(x)/(2))`
`= (2)/(4) underset(x to 0)(lim) (sin(x)/(2))/((x)/(2)) underset(x to 0)(lim) (sin(X)/(2))/((x)/(2)) = (1)/(2) 1.1 = (1)/(2)`
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Knowledge Check

  • lim_(x to 0) (1- cos 2x)/x

    A
    0
    B
    1
    C
    2
    D
    4

  • lim_(x to 0) (1 - cos x)/(x sqrt(x^(2))

    A
    `(1)/(2)`
    B
    `-(1)/(2)`
    C
    0
    D
    Does not exist

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