3Solve Find the Limit Homework Statement

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In summary, the conversation discusses finding the limit of (sin(3x))/x as x approaches 0. The steps involve using a substitution and a geometrical argument to prove that the limit is equal to 1. Other methods, such as using the del'hopitale rule or the Taylor expansion of sine, can also be used to verify the limit. The speaker expresses a desire to understand the reasoning behind the solution and mentions having an obsessive-compulsive tendency towards learning.
  • #1
brycenrg
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Homework Statement


Find the limit
lim (sin(3x))/x
x-->0


Homework Equations


There is two steps which I don't understand how they work, or why they work. Ill show you.
I'm using my answer book to help me out, but it doesn't give a satisfactory explanation.


The Attempt at a Solution


lim (sin(3x))/x
x-->0

= lim 3(sin(3x))/3x
x-->0

= 3lim (sin(3x))/3x as x -> 0 3x -> 0 Why do I do this maneuver? I understand its for
x-->0 substitution

= 3lim (sin(3x))/3x let theta = 3x

= 3lim (sin(theta))/theta

= 3(1) Why does lim (sin(theta))/theta equal 1?
 
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  • #2
brycenrg said:
Why does lim (sin(theta))/theta equal 1?

Here's a geometrical argument that 'proves' the limit is equal to 1: http://math.ucsd.edu/~wgarner/math20a/sin%28x%29_over_x.htm [Broken] .

I think this argument is enough for your proof, you're probably not supposed to deduce the limit from the Maclaurin series expansion of the sine function (which would be the rigorous way to do it).
 
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  • #3
There are many ways to check it... first of all you have a form of 0/0 for which you can use the del'hopitale...
Another way is the Taylor expansion of sine : the argument is small so you just need the first order term which is [itex]\theta[/itex] and gives 1 with the denominator...
Another way is via the sandwich rule demonstrated by hilbert12's reference.
 
  • #4
Thank you very much. I like to learn how it works for some odd reason, ocd or something. Its like i doubt maths even though its been around for ever lol. Thanks.
 

1. What is "3Solve Find the Limit Homework Statement"?

3Solve Find the Limit Homework Statement is a type of homework assignment commonly given in math and science classes. It involves finding the limit of a function as its input approaches a certain value or infinity.

2. Why is finding limits important?

Finding limits is important because it allows us to understand the behavior of a function as its input approaches a certain value. This can help us make predictions and solve problems in various areas of math and science.

3. How do you solve a "3Solve Find the Limit Homework Statement" problem?

The specific steps for solving a "3Solve Find the Limit Homework Statement" problem may vary, but generally you will need to evaluate the function at the given input value and see if it approaches a specific value or infinity. You may also need to use algebraic techniques, such as factoring or simplifying, to manipulate the function and find the limit.

4. What are some common strategies for finding limits?

Some common strategies for finding limits include direct substitution, factoring, simplifying, using trigonometric identities, and applying L'Hopital's rule. It is also helpful to know the properties of limits, such as the sum, difference, product, and quotient rules.

5. What are some common mistakes to avoid when solving a "3Solve Find the Limit Homework Statement" problem?

Some common mistakes to avoid when solving a "3Solve Find the Limit Homework Statement" problem include forgetting to check for undefined values, using incorrect algebraic manipulations, and not understanding the properties of limits. It is also important to carefully evaluate the function at the given input value and not assume that it approaches a certain value without proper justification.

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