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mtayab1994
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Homework Statement
find the limit of:
[tex]\lim_{x\rightarrow0}\frac{1-cos(2x)}{tan(x)}[/tex]
The Attempt at a Solution
[tex]\lim_{x\rightarrow0}\frac{1-cos(2x)}{tan(x)}=\frac{1-1-\frac{x^{2}}{2}}{tan(x)}=0[/tex]
That should bemtayab1994 said:Homework Statement
find the limit of:
[tex]\lim_{x\rightarrow0}\frac{1-cos(2x)}{tan(x)}[/tex]
The Attempt at a Solution
[tex]\lim_{x\rightarrow0}\frac{1-cos(2x)}{tan(x)}=\frac{1-1-\frac{x^{2}}{2}}{tan(x)}=0[/tex]
The lim symbol should remain until you actually take the limit. In the expression in the middle, above, the denominator is 0, so you can't conclude that the limit is zero.mtayab1994 said:Homework Statement
find the limit of:
[tex]\lim_{x\rightarrow0}\frac{1-cos(2x)}{tan(x)}[/tex]
The Attempt at a Solution
[tex]\lim_{x\rightarrow0}\frac{1-cos(2x)}{tan(x)}=\frac{1-1-\frac{x^{2}}{2}}{tan(x)}=0[/tex]
See above.kushan said:you are doing right
This is definitely the way to go.SammyS said:That should be[itex]\displaystyle \frac{1-\cos(2x)}{\tan(x)}=\frac{1-\left(1-\frac{(2x)^{2}}{2}+\dots\right)}{\tan(x)}=\frac{1-1+\frac{(2x)^{2}}{2}-\dots}{\tan(x)}[/itex]
You can do this without using L'Hôpital's rule or Taylor series.
Start with cos(2x) = 1 - 2sin2(x) and 1/tan(x) = cos(x)/sin(x) .
Mark44 said:This is definitely the way to go.
No. How did you get from (1 - cos(2x))/tan(x) to (-sin2(x)cos(x))/sin(x)? Show us what you did to get that.mtayab1994 said:after doing that i got:
[tex]\lim_{x\rightarrow0}\frac{-sin^{2}(x)*cos(x)}{sin(x)}=-sin(x)*cos(x)=-sin(0)*cos(0)=0*1=0[/tex]
Is that correct?
Mark44 said:No. How did you get from (1 - cos(2x))/tan(x) to (-sin2(x)cos(x))/sin(x)? Show us what you did to get that.
Also, the lim symbol needs to appear until you actually take the limit. In other words, as long as x is still in any expression, you haven't takent the limit.
You have two mistakes. That should be 1 - (1 - 2sin2(x)).mtayab1994 said:Well, i replaces 1-cos(2x) with 1-1-sin^2(x) and the bottom from tan(x) to sin(x)/cos(x)
mtayab1994 said:Then i removed the denominator by 1/tan(x).
Yes.mtayab1994 said:alright and for the denominator should i do tanx=sinx/cosx?
2sin(x)cos(x) doesn't equal 0, but the limit as x approaches 0 of this expression is 0.mtayab1994 said:Ok i got (2sin^2(x)*cos(x))/(sin(x))
then i canceled out with the sine and i was left with 2sinx*cosx which then equals 0 is that correct?
Mark44 said:Yes.
2sin(x)cos(x) doesn't equal 0, but the limit as x approaches 0 of this expression is 0.
In other words,
2sin(x)cos(x) ≠ 0, but [itex]\lim_{x \to 0} 2sin(x)cos(x) = 0[/itex]. There's a difference here.
The value of the limit is 0, but what you wrote is not correct.mtayab1994 said:Yea sorry i didn't right that but is what i did correct?
Again, 2sinx * cosx ≠ 0, in general.mtayab1994 said:i was left with 2sinx*cosx which then equals 0
Mark44 said:The value of the limit is 0, but what you wrote is not correct.
Again, 2sinx * cosx ≠ 0, in general.
Yes, but what you said didn't include the word "limit," so isn't true.mtayab1994 said:I don't get it what's wrong with that. Isn't the limit as x approaches 0 of 2sinx*cosx=0?
Mark44 said:Yes, but what you said didn't include the word "limit," so isn't true.
True:
[tex]\lim_{x \to 0}~2sin(x)cos(x) = 0[/tex]
False:
2sin(x)cos(x) = 0
I'm am trying to get you to distinguish between the value of an expression and the value of the limit of that expression.
Do you see the difference now?
A limit is a mathematical concept that represents the value that a function or sequence approaches as the input or index approaches a certain value. It is used to describe the behavior of a function or sequence as the input or index becomes infinitely large or infinitely small.
To find the limit of a function, you can either use algebraic techniques such as factoring, simplifying, and canceling, or you can use graphical techniques such as looking at the behavior of the function on a graph or using a table of values. You can also use calculus techniques such as differentiation and integration to find limits.
The purpose of checking a limit is to understand the behavior of a function or sequence as the input or index approaches a certain value. It can help us determine the continuity, differentiability, and convergence of a function or sequence, and it is also essential in many applications in physics, engineering, and economics.
Yes, a limit can sometimes not exist. This can happen when the function has a vertical asymptote, when the right-hand and left-hand limits approach different values, or when the function oscillates between two values as the input approaches a certain value. In these cases, we say that the limit does not exist.
Checking a limit is important because it allows us to understand the behavior of a function or sequence in different scenarios. It helps us determine the continuity and differentiability of a function, which are crucial concepts in calculus. It also helps us make predictions and solve problems in various fields such as physics, engineering, and economics.