Finding a limit using substitution rule, my answer is 0, my book`s is -2

So, you'll have to say something like "If \cos(x)\geq0 then ..." In summary, we have a limit of 0 as x approaches pi. The book mentions -2 as the answer, but this is because of a mistake. In order to solve this, we can substitute u=cos(x) and use the fact that \sec(x)=1/\sqrt{1-y}. However, there is an issue with the sign of sec(x) depending on the value of cos(x), so we have to consider two cases. Ultimately, we will still get a limit of 0 as x approaches pi.
  • #1
wajed
57
0

Homework Statement



Lim [(tanx)^2] / [1 + secx] <<< as x goes to pi



Homework Equations





The Attempt at a Solution



(tan x)^2 = (sin x)^2 / (cos x)^2

(sin x)^2 = y

lim y = 0 <<< as x goes to pi


lim [y/ (cos x)^2] / [1 + (1/y)] <<< as y goes to C=0

1+ (1/y) = (y+1)/y

lim [y^2] / (y+1) (cos x)^2

y=0

so, 0/(0+1)(cos 0)^2 = 0/1(1) = 0/1 = 0

why does my book mentions that answer is -2?
 
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  • #2
huh, that was a stupid mistake, sorry.
(I`m just totally nervous, I got an exam tomorrow)EDIT:
but, how do I solve that anyway?

I can`t manage to do it..

I know the whole thing is about the "cos x" and that I have to make it turn to something in terms of y, but how?
 
  • #3
sec(x) is not 1/sin(x). It's 1/cos(x). Try the substitution u = cos(x). :smile:
 
  • #4
wajed said:
lim [y/ (cos x)^2] / [1 + (1/y)]

Not quite. If [itex]\sin^2(x)=y[/itex] then [itex]\cos^2(x)=1-y[/itex]. Here's the tricky part: [itex]\sec(x)=1/\sqrt{1-y}[/itex] if [itex]\cos(x)\geq0[/itex] but [itex]\sec(x)=-1/\sqrt{1-y}[/itex] if [itex]\cos(x)<0[/itex].
 

Related to Finding a limit using substitution rule, my answer is 0, my book`s is -2

1. What is the substitution rule for finding a limit?

The substitution rule for finding a limit states that if the limit of a function f(x) as x approaches a is equal to L, then the limit of f(g(x)) as x approaches a is also equal to L, where g(x) is a function that approaches a as x approaches a.

2. How do I know when to use the substitution rule to find a limit?

You can use the substitution rule to find a limit when you have a rational function or a function that can be simplified to a rational function, and the limit is indeterminate (e.g. 0/0 or ∞/∞).

3. Why is the answer to my substitution rule limit problem 0, but my book's answer is -2?

The answer to a substitution rule limit problem can vary depending on the function and the value of a. It is possible that you made a mistake in your calculations or that the book's answer is incorrect. It is always a good idea to double check your work and consult with your instructor if you are unsure.

4. Can I use the substitution rule to find limits at infinity?

Yes, you can use the substitution rule to find limits at infinity. In this case, you would substitute x with 1/t (or another appropriate expression) and take the limit as t approaches 0.

5. Is the substitution rule the only method for finding limits?

No, the substitution rule is not the only method for finding limits. There are other methods such as direct substitution, factoring, rationalizing, and using L'Hopital's rule. It is important to use the method that is most appropriate for the given function and limit problem.

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