We are doing limits in Calculus, but now we are doing trig limits too and i do not get how to do any of them.
For example ones like this, how do you do them?
sec (x-1) / (x sec x) x approaches 0
((3 sin (x)) 1 - cos(x)) / (x^2) x approaches 0
(sin (2x)) / (sin (3x)) x approaches 0
(1-tan(x)) / (sin (x) - cos (x)) x approaches 0
I just want to learn how to these kinds of limits so i know how to do all the types.
Please help, is there a website that will teach me how to do these kinds of limits?
Thank You.
If going to the limit results in a 0/0 or infinity/innfinity indeterminate fraction, then take the ratio of the derivatives of the numerator and denominator to get the limit. That is called L'Hopital's rule.
For the question
Lim (sin (2x)) / (sin (3x))
x->0
the answer is [2 (cos(0)]/[3 cos(0)] = 2/3
Remember that sin x is very close to x when x is small. this tells you right away that it approaches 2x/3x = 2/3
For a website that should help, Google "L'Hopital's rule".
One good site is
http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule
Trigonometric limits can sometimes be challenging, but with practice and understanding of some important trigonometric identities, you can become comfortable solving them. I'll explain the general approach to solving trigonometric limits, and then provide you with some steps to tackle the specific examples you mentioned.
When dealing with trigonometric limits, it is important to remember some essential trigonometric identities:
1. sin(2x) = 2sin(x)cos(x)
2. cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x) = 2cos^2(x) - 1
3. tan(x) = sin(x) / cos(x)
4. sec(x) = 1 / cos(x)
Now let's take a look at the examples you provided:
1. lim[x→0] sec(x-1) / (x sec(x))
To start, let's simplify this expression using the trigonometric identity sec(x) = 1 / cos(x):
lim[x→0] (1 / cos(x-1)) / (x / cos(x))
Now, multiply the numerator and denominator by cos(x) to get rid of the fraction:
lim[x→0] (cos(x) / cos(x-1)) / (x / cos(x))
Now, the limit is in an indeterminate form of 0/0. To resolve this, we can use L'Hôpital's Rule, which states that if the limit of the ratio of derivatives exists, it is equal to the limit of the original function:
lim[x→0] (cos(x) / cos(x-1)) / (x / cos(x))
= lim[x→0] [(d/dx)cos(x) / (d/dx)cos(x-1)] / [(d/dx)x / (d/dx)cos(x)]
Differentiating the numerator and denominator, we get:
lim[x→0] [-sin(x) / sin(x-1)] / [1 / sin(x)]
= lim[x→0] [-sin(x)sin(x) / sin(x-1)]
= -sin(0)sin(0) / sin(0-1)
= 0
Therefore, the limit is 0.
2. lim[x→0] ((3sin(x)(1-cos(x))) / (x^2))
In this example, we can simplify the expression using the trigonometric identity 1 - cos(x) = 2sin^2(x/2):
lim[x→0] ((3sin(x)(1-cos(x))) / (x^2))
= lim[x→0] ((3sin(x)(2sin^2(x/2))) / (x^2))
= lim[x→0] (6sin^3(x/2) / (x^2))
Now, we have a sin^3 and x^2, which are both indeterminate forms. To resolve this, we can again use L'Hôpital's Rule:
lim[x→0] (6sin^3(x/2) / (x^2))
= lim[x→0] [(d/dx)6sin^3(x/2) / (d/dx)x^2]
Differentiating the numerator and denominator, we get:
lim[x→0] [(d/dx)6sin^3(x/2) / (d/dx)x^2]
= lim[x→0] [6(3sin^2(x/2)(d/dx)sin(x/2)) / (2x)]
= lim[x→0] [6(3sin^2(x/2)(cos(x/2)/2)) / (2x)]
= [6(3(0)(1/2)) / (0)]
= 0
Therefore, the limit is 0.
3. lim[x→0] (sin(2x) / sin(3x))
To solve this limit, we can use the trigonometric identity sin(2x) = 2sin(x)cos(x):
lim[x→0] (2sin(x)cos(x) / sin(3x))
= lim[x→0] (2sin(x)cos(x) / 3sin(x)cos^2(x))
= lim[x→0] (2/3cos(x))
= 2/3
Therefore, the limit is 2/3.
4. lim[x→0] (1-tan(x)) / (sin(x)-cos(x))
To start, we can replace tan(x) using the identity tan(x) = sin(x) / cos(x):
lim[x→0] (1 - sin(x) / cos(x)) / (sin(x) - cos(x))
= lim[x→0] ((cos(x) - sin(x)) / cos(x)) / ((sin(x) - cos(x)))
Now, notice that the numerator is equal to -(sin(x) - cos(x)), so we can rewrite the expression as:
lim[x→0] -1 / cos(x)
= -1 / cos(0)
= -1
Therefore, the limit is -1.
Now, regarding your question about websites that can help you learn how to solve trigonometric limits, there are several online resources available. Some recommended websites include Khan Academy (www.khanacademy.org) and Paul's Online Math Notes (tutorial.math.lamar.edu). These websites offer comprehensive explanations, examples, and practice problems to help you master trigonometric limits in Calculus.
Remember that practice is essential to becoming proficient in solving trigonometric limits. As you solve more problems and gain familiarity with the various trigonometric identities, you'll become more comfortable in handling these types of limits.