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Math Help - Help with Trig Limits

  1. #1
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    Help with Trig Limits

    I understand the basicsof trig limits, but of a gamut of 500 questions there are several I just can't get the answer for. What am I doing wrong?

    1. Lim (sinx / x+tanx)
    x->0

    2. Lim([cot^2]x / 1-cscx)
    x->pi/2

    3. Lim (tan4x/tan7x)
    x->0

    1. I tend to end up at a mess here: I multiply the top and bottom by 'x' so I can use the sinx/x identity, and ultimately end up with (xcosx/(xcosx+sinx)) and get lost from there. According to the back of the book, the answer is 1/2.

    2. Everything I try here just ends up a mess

    3. I tend to end up at

    (sin4x)/4x . 7x/(sin7x) . 4x/(cos4x) . (cos7x/7x)
    =(4/7) . (0) . (undefined)
    the answer in the back of the book is 4/7,and I don't know how to get there

    Thanks in advance! I have a test Tuesday,and I'm so lost in all of this.
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  2. #2
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    Quote Originally Posted by Futant View Post
    I understand the basicsof trig limits, but of a gamut of 500 questions there are several I just can't get the answer for. What am I doing wrong?

    1. Lim (sinx / x+tanx)
    x->0

    2. Lim([cot^2]x / 1-cscx)
    x->pi/2

    3. Lim (tan4x/tan7x)
    x->0

    1. I tend to end up at a mess here: I multiply the top and bottom by 'x' so I can use the sinx/x identity, and ultimately end up with (xcosx/(xcosx+sinx)) and get lost from there. According to the back of the book, the answer is 1/2.

    2. Everything I try here just ends up a mess

    3. I tend to end up at

    (sin4x)/4x . 7x/(sin7x) . 4x/(cos4x) . (cos7x/7x)
    =(4/7) . (0) . (undefined)
    the answer in the back of the book is 4/7,and I don't know how to get there

    Thanks in advance! I have a test Tuesday,and I'm so lost in all of this.
    for 1 )
    \frac{\sin(x)}{x+\tan(x)}=\frac{\sin(x)}{x+\frac{\  sin(x)}{\cos(x)}}=\frac{\sin(x)}{\sin(x)}\cdot \frac{1}{\frac{x}{\sin(x)}+\frac{1}{\cos(x)}} Now as x \to 0 we get \frac{1}{1+1}=\frac{1}{2}

    for 2) use the identity that 1+\cot^2(x)=csc^2(x) and factor and reduce.

    for 3) multiply the top and bottom by \frac{4\cdot 7 \cdot x}{4 \cdot 7 \cdot x} and do what you tried. (you have an extra 4x)
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