Limit question

  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. gb7nash

    gb7nash 805
    Homework Helper

    The bolded part is correct. However, what is cos(x) approaching as x approaches 0?
  4. cos(x) approaches 1..
  5. gb7nash

    gb7nash 805
    Homework Helper

    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. Thank you.
  7. I wasn't aware that you can't use that rule if the function actually approaches a value.
  8. gb7nash

    gb7nash 805
    Homework Helper

    [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. that makes sense. Thank you.
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