Solve Trig Limit Problem: lim (x->0) \frac{x - sinx}{x^{3}}

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In summary, a trigonometric limit is a mathematical concept that describes the behavior of a function as the input variable approaches a specific value, and the general approach to solving trigonometric limit problems involves simplifying the expression and evaluating the limit using algebraic manipulation and trigonometric identities. The specific approach to solving the limit problem lim (x->0) \frac{x - sinx}{x^{3}} is to use a trigonometric identity, cancel common factors, and apply limit laws. Some common mistakes to avoid when solving trigonometric limit problems include forgetting to use trigonometric identities and incorrectly applying limit laws. Understanding trigonometric limits can be useful in real-world applications such as physics, engineering, and statistics.
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rambo5330
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Homework Statement



is there a way to solve

lim (x->0) [tex]\frac{x - sinx}{x^{3}}[/tex]


using the fact that sinx / x = 1

or is it much more complicated.. I've tried to break it down into sinx/x everyway i can think of with no luck..
 
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  • #2
Do you know l'Hopitals rule?

I am thinking of a way to do it with only sin(x)/x but can't come up with a good way immediately.
 
  • #3
I am not familiar with l'hopitals theorem yet...
 

Related to Solve Trig Limit Problem: lim (x->0) \frac{x - sinx}{x^{3}}

What is the definition of a trigonometric limit?

A trigonometric limit is a mathematical concept that describes the behavior of a function as the input variable approaches a specific value, typically infinity or a particular point on the unit circle. In other words, it describes how the values of a trigonometric function change as its input gets closer to a certain value.

What is the general approach to solving trigonometric limit problems?

The general approach to solving trigonometric limit problems is to use algebraic manipulation, trigonometric identities, and limit laws to simplify the expression and then evaluate the limit. This may involve rewriting the expression in a different form, using trigonometric identities to eliminate complex terms, or factoring and canceling common factors.

What is the specific approach to solving the trigonometric limit problem lim (x->0) \frac{x - sinx}{x^{3}}?

The specific approach to solving this trigonometric limit problem is to use the trigonometric identity sinx = x - (1/2)x^3 + O(x^5) to rewrite the expression as lim (x->0) \frac{x - (x - (1/2)x^3 + O(x^5))}{x^3}. Then, we can cancel the x terms and use the limit laws to simplify the expression to lim (x->0) \frac{1/2x^3 + O(x^5)}{x^3}. Finally, we can apply the limit laws again to evaluate the limit and get the answer of 1/2.

What are some common mistakes to avoid when solving trigonometric limit problems?

Some common mistakes to avoid when solving trigonometric limit problems include forgetting to use trigonometric identities, not simplifying the expression enough before evaluating the limit, and incorrectly applying limit laws. It is also important to pay attention to the specific conditions of the limit, such as whether it is a one-sided or two-sided limit.

How can understanding trigonometric limits be useful in real-world applications?

Understanding trigonometric limits can be useful in real-world applications such as physics, engineering, and statistics. For example, in physics, trigonometric limits can help us understand the behavior of waves, while in engineering, they can be used to optimize the design of structures. In statistics, trigonometric limits can be used to analyze and predict patterns in data.

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