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Determine: 1.1 lim (x->1) (x/ln x - 1/ln x) 1.2 lim (x->inf) x sin (1/x) 2 If f(x) = (2x)/(1-sin 2x) Find the value(s) of x that will make the function f(x) to be discontinuous for x ∈ [0°; 90°]

Mathematics

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Answer

1\text{1}

Graph of f(x) = (2x)/(1 − sin(2x))

Use the left and right arrow keys to trace the curve, up and down to jump between key points, and plus and minus to zoom.

Key points of f(x) = (2x)/(1 − sin(2x))
Pointxy
Maximum−3.8−3.86
x-intercept00
Minimum2.132.24
Minimum5.415.45

Step-by-step solution

3 steps

  1. Combine the fractions.

    lim⁡x→1(xln⁡x−1ln⁡x)=lim⁡x→1x−1ln⁡x\lim_{x \to 1} \left( \frac{x}{\ln x} - \frac{1}{\ln x} \right) = \lim_{x \to 1} \frac{x-1}{\ln x}

  2. Evaluate the limit.

    Substituting x=1x=1 gives 1−1ln⁡1=00\frac{1-1}{\ln 1} = \frac{0}{0}, which is an indeterminate form. Apply L'Hôpital's Rule.

    lim⁡x→1ddx(x−1)ddx(ln⁡x)=lim⁡x→111x\lim_{x \to 1} \frac{\frac{d}{dx}(x-1)}{\frac{d}{dx}(\ln x)} = \lim_{x \to 1} \frac{1}{\frac{1}{x}}

  3. Simplify and substitute x=1x=1.

    lim⁡x→1x=1\lim_{x \to 1} x = 1

    The final answer is 1\text{1}.

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