Evaluating Limits with trig functions

In summary, the conversation discusses how to evaluate the limit of sin(pi/x) sqrt(x^3+x^2) as x approaches 0. It is suggested to use the fact that the absolute value of sine is always less than or equal to 1. Further explanations are given on how to use this knowledge to solve similar problems involving trig functions.
  • #1
PhysChem
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0

Homework Statement


lim x-->0 sin(pi/x) sqrt(x^3+x^2)

The Attempt at a Solution



I was having trouble evaluating the above limit. Do I start by isolating x? For some reason, when it comes to trig functions such as this, I'm not sure how to simplify it. Also, what material would I have to review for me to understand how to break down such trig functions? I'm aware of fundamental, quotient and reciprocal identities of trig functions but am not sure how to use that knowledge to solve these type of problems.
 
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  • #2
As x goes to 0, sin(pi/x) oscillates but stays between -1 and 1. That is,
[tex]-\sqrt{x^3+ x^2}\le sin(\pi/x)\sqrt{x^3+ x^2}\le \sqrt{x^3+ x^2}[/tex]
 
  • #3
Use the fact that :

|sin(x)| ≤ 1 [itex]\forall[/itex]x[itex]\ni[/itex]R

As in FOR ALL x you happen to plug into the sin function including whatever value gets spat out of pi/x.

hint hint ;)
 

Related to Evaluating Limits with trig functions

1. What is a limit in mathematics?

A limit in mathematics is the value that a function approaches as the input variable gets closer and closer to a specific value. It is used to describe the behavior of a function and determine its value at a specific point.

2. How do trigonometric functions affect limits?

Trigonometric functions, such as sine, cosine, and tangent, can be used in limits to evaluate the behavior of a function as it approaches a specific value. They can also be used to find the exact value of a limit in some cases.

3. What are the common strategies for evaluating limits with trig functions?

The most common strategies for evaluating limits with trig functions include using trigonometric identities, factoring, and rewriting the function in terms of sine and cosine. You can also use the squeeze theorem and L'Hopital's rule in some cases.

4. Can limits with trig functions have multiple answers?

Yes, depending on the function and the value it approaches, a limit with trig functions can have multiple answers. This is because trigonometric functions have periodic behavior, and the value of the limit may repeat at different intervals.

5. Why is it important to evaluate limits with trig functions?

Evaluating limits with trig functions is important because it allows us to understand the behavior of a function and determine its value at a specific point. It is also a fundamental concept in calculus and is used in many real-world applications, such as in physics and engineering.

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