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Math Help - limits question

  1. #1
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    Angry limits question

    Let f (x) be a function defined for all x, with −5 ≤ f (x) ≤ 10. Also, lim f(x) as x ->0 does not exist but f (0) = 3.

    (a) Let g (x) = xf(x). Show g is continuous at x = 0.
    i know that for the above i have to use the 3 conditions
    f(c) exists
    lim f(x) as x->0- = lim f(x) as x -> 0+
    lim f(x) as x -> 0 = f(c)

    (b) Does the graph of g have a tangent line at (0, 0)? Explain.

    for this one i'm not sure because g(x)= xf(x)
    so g'(x) = f(x) + xf'(x)
    g'(x) = 3 + 0 * f'(x) [ i'm not sure about the value of f'(x)
    can anyone help me with the value of f'(x) and whether g(0) has a tangent line
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  2. #2
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    Re: limits question

    Quote Originally Posted by ubhutto View Post
    (a) Let g (x) = xf(x). Show g is continuous at x = 0.
    i know that for the above i have to use the 3 conditions
    f(c) exists
    lim f(x) as x->0- = lim f(x) as x -> 0+
    lim f(x) as x -> 0 = f(c)
    I assume the three conditions should be formulated for g and not f. Obviously, g(0) = 0. For the other two conditions, it is easier to apply the ε-δ definition of continuity. Suppose you are given an ε > 0. Which δ > 0 can you choose so that |x - 0| < δ would guarantee that |g(x) - g(0)| < ε? Note that |g(x) - g(0)| = |x * f(x)| = |x| * |f(x)| and |f(x)| ≤ 10.

    Quote Originally Posted by ubhutto View Post
    (b) Does the graph of g have a tangent line at (0, 0)? Explain.

    for this one i'm not sure because g(x)= xf(x)
    so g'(x) = f(x) + xf'(x)
    g'(x) = 3 + 0 * f'(x) [ i'm not sure about the value of f'(x)
    can anyone help me with the value of f'(x) and whether g(0) has a tangent line
    You can only apply the product rule when both functions are differentiable at the given point. Here f(x) is not even continuous at 0. Try using the definition of derivative as a limit.
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