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Limit question

  1. Oct 9, 2011 #1
    Problem:

    Lim sin(cos(x))/sec(x)
    x -> 0

    The answer in the book is sin(1).. which is obvious... but why do I not arrive at the same answer doing is this way.....

    Knowing as x approaches zero, sin(x)/x approaches 1....

    ( Sin(cos(x))/cos(x) )(cos^2(x)) = (1)(1) = 1 = / = sin(1)...???
     
  2. jcsd
  3. Oct 9, 2011 #2

    gb7nash

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    The bolded part is correct. However, what is cos(x) approaching as x approaches 0?
     
  4. Oct 9, 2011 #3
    cos(x) approaches 1..
     
  5. Oct 9, 2011 #4

    gb7nash

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    Ok, so since cos(x) goes to 1:

    [tex]\lim_{x \to 0} \frac{\sin(\cos(x))}{\cos(x)}[/tex]

    is equivalent to:

    [tex]\lim_{u \to 1} \frac{\sin(u)}{u}[/tex]

    We can't apply the sin(_)/_ rule here. The term inside of the sine approaches 1.
     
  6. Oct 9, 2011 #5
    Thank you.
     
  7. Oct 9, 2011 #6
    I wasn't aware that you can't use that rule if the function actually approaches a value.
     
  8. Oct 9, 2011 #7

    gb7nash

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    [tex]\lim_{x \to 0}\frac{\sin(x)}{x} = 1[/tex]

    x needs to approach 0. If it approaches any other value (or does not exist), you can't use this identity.
     
  9. Oct 9, 2011 #8
    that makes sense. Thank you.
     
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