Exploring the Limit of (sin2x)^4/x^4: A Trigonometric Challenge

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In summary, the problem is to find the limit or prove it does not exist for the function (sin2s)^4/s^4 as s approaches 0. The student is having trouble manipulating the function into the form sinx/x = 1 and is considering taking the 2 out as a constant. However, they are unsure how to handle the difference between (sinx)^4 and sin(x^4). Another student suggests using the identity sin(ax)/(ax) = 1 and the problem is eventually solved with the final answer being 16.
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beneakin
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Homework Statement


Find the limit or prove it does not exist.

lim as s->0 (sin2s)^4/s^4


Homework Equations


i know i have to use sinx/x = 1 but I am having trouble manipulating the function into that form


The Attempt at a Solution



my first thought was that i could just take the 2 out as a constant would that just be (2^4)(sinx)^4/x^4

but then i have (sinx)^4 not sinX^4 I am stuck on what king of algebra will work here
 
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  • #2
a^2/b^2=(a/b)^2
 
  • #3
Lim_(x->0) of sin(ax)/(ax)=1
 
  • #4
ahh so then (2*1)^4 = 16

thanks alot
 
  • #5
beneakin said:
ahh so then (2*1)^4 = 16

thanks alot
good, welcome!
 

1. What is a limit in calculus?

A limit is a fundamental concept in calculus that represents the value that a function approaches as its input (x) approaches a certain value. It is often used to describe the behavior or trend of a function near a specific point.

2. How do you find a limit with trigonometric functions?

To find a limit with trigonometric functions, you can use algebraic techniques such as factoring, rationalizing the numerator or denominator, and simplifying trigonometric expressions. You can also use trigonometric identities and properties to manipulate the function and then evaluate the limit.

3. Can you use the squeeze theorem to find a limit with trig?

Yes, the squeeze theorem can be used to find a limit with trigonometric functions. This theorem states that if two functions, f(x) and g(x), have the same limit as x approaches a certain value, and there is a third function, h(x), that is always between f(x) and g(x) for values of x near the limit, then h(x) also has the same limit.

4. How do you handle undefined values when finding a limit with trig?

Undefined values, such as division by zero, can make it difficult to evaluate limits with trigonometric functions. In these cases, you can try to simplify the function or use trigonometric identities to rewrite the function in a form that does not have an undefined value. If this is not possible, the limit may not exist or you may need to use more advanced techniques, such as L'Hôpital's rule.

5. What are some common trigonometric limits?

Some common trigonometric limits include:
- sin(x)/x as x approaches 0
- cos(x)/x as x approaches 0
- tan(x)/x as x approaches 0
- sin(x)/x as x approaches infinity
- cos(x)/x as x approaches infinity
- tan(x)/x as x approaches infinity
- sin(x)/x as x approaches a constant value (using the Squeeze Theorem)
- cos(x)/x as x approaches a constant value (using the Squeeze Theorem)
- tan(x)/x as x approaches a constant value (using the Squeeze Theorem)

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