Positive integers for k: finding limits

In summary, the limit as x approaches 0 of sin(sin(x))/x^k exists for k = 1 and is equal to 1. For k > 2, the limit is undefined and for k = 2, it is infinity. The existence of the limit for other positive values of k cannot be determined based on the given information.
  • #1
chapsticks
38
0

Homework Statement


find the positive integers k for which

lim x->0 sin(sin(x))/x^k

Homework Equations



exists, and then find the value of the limit

The Attempt at a Solution


I did the first three k's

k=0
lim x->0 sin(sin x))/x^0= 0 undefined

k=1
lim x->0 sin(sinx))/x^1= 1

k=2 I might be wrong with this
lim x->0 sin(sinx))/x^2= undefined

k>2 how do I do this one.
lim x-> sin(sinx))/x^k>2=?
 
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  • #2
0 is not positive.

Regarding k = 2: How did you arrive at your answer for k = 1 ??
 
  • #3
well I plugged it into my calculator
 
  • #4
What happens if you multiply and divide sin(x)?
 
  • #5
I'm not sure.
 
  • #6
chapsticks said:
I'm not sure.

Do it an see what you recognize
 
  • #7
well in the table for k=2

it goes from -100 then an error then 99

so it must be for all positive numbers and above should be infinity?
 
  • #8
chapsticks said:
well I plugged it into my calculator

Are you saying that your calculator says that sin(sin(0))/0 = 1 ?
 
  • #9
I used a different method.. but that answer sin(sin(0))/0 does not exist right
 
  • #10
chapsticks said:
I used a different method.. but that answer sin(sin(0))/0 does not exist right
Right.

So, I ask again. How did you find the answer when k = 1 ?
 

What are positive integers?

Positive integers are whole numbers that are greater than zero. They do not include any decimal or fractional parts.

What is the purpose of finding limits for positive integers?

The purpose of finding limits for positive integers is to determine the behavior of a function as the input values approach a certain value. This can help in understanding the overall behavior and trends of a function.

How do you find the limits for positive integers?

To find the limits for positive integers, you can use algebraic methods such as direct substitution, factoring, or using limit laws. You can also use graphical methods or numerical methods such as tables or calculators.

What are the key properties of limits for positive integers?

The key properties of limits for positive integers include the limit laws, which state that the limit of a sum, difference, product, or quotient of two functions is equal to the sum, difference, product, or quotient of their individual limits. Another important property is the squeeze theorem, which helps to determine the limit of a function by comparing it with other known functions.

Why are limits important in mathematics and science?

Limits are important in mathematics and science because they help to define and understand the behavior of functions, which are essential in many real-life applications and models. They also allow us to make predictions and approximations, which are crucial in problem-solving and decision-making.

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