Now, can you prove this limit using the Limit Definition of Continuity

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In summary, the Limit Definition of Continuity is a mathematical concept that defines the continuity of a function at a specific point. It is used to prove the continuity of a function by evaluating its limit at that point. This definition can be applied to all types of functions and is different from the traditional definition of continuity as it only considers the first two criteria. An example of using this definition to prove continuity is demonstrated with the function f(x) = x^2 at x = 2, where the limit is equal to the function's value at that point.
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Jameson
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**Note - this for young students who might be studying pre-calculus or just beginning calculus and haven't been presented with the concept of upper/lower bounds and how they are useful in working with trig expressions.**

Consider \(\displaystyle \lim_{x \rightarrow 0} x^4 \cos \left( \frac{1}{x^2} \right)\).

You can easily graph this and visually deduce that the answer is 0, which it is. Along the same lines you could make a table of (x,y) points and choose x's that get closer and closer to x=0 and again conclude that the limit is most likely 0. Although these two techniques might be useful on a test where you're stuck on a tricky limit, the answer can be found in a far more certain way.

So this week's question is prove that the above limit is 0.

Hint:
[sp]Start with the fact that \(\displaystyle -1 \le \cos(x) \le 1\) with the goal of manipulating this somehow to where the middle term becomes the limit we want to find. Note that in the above, $\cos(x)$ cannot be smaller than -1 or larger than 1 for any value of x. This is true even if "x" is something more complicated. If you can't see how this applies in this problem, see hint 2.[/sp]

Hint 2:
[sp]\(\displaystyle -1 \le \cos \left( \frac{1}{x^2} \right) \le 1\) Now try manipulating this but make sure to do the same operation to all of the terms so it remains true, and remember that inequalities are not the same as equations and you must always keep positives and negatives in mind. [/sp]
 
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Congratulations to the following members for their correct solutions:

1) Sudharaka

Solution (from Sudharaka):

[sp] \[-1 \le \cos \left( \frac{1}{x^2} \right) \le 1\]

Since \(x^4\geq 0\) we have,

\[\Rightarrow -x^4\le x^4 \cos \left( \frac{1}{x^2} \right) \le x^4\]

When \(x\rightarrow 0\) by the Squeeze Theorem (Squeeze theorem - Wikipedia, the free encyclopedia) we get,

\[\lim_{x \rightarrow 0} x^4 \cos \left( \frac{1}{x^2} \right)=0\][/sp]
 

FAQ: Now, can you prove this limit using the Limit Definition of Continuity

1. What is the Limit Definition of Continuity?

The Limit Definition of Continuity is a mathematical concept that defines the continuity of a function at a specific point. It states that a function is continuous at a point if the limit of the function at that point exists, and is equal to the value of the function at that point. In other words, a function is continuous if its graph has no breaks or gaps at the given point.

2. How is the Limit Definition of Continuity used?

The Limit Definition of Continuity is used to mathematically prove the continuity of a function at a given point. By evaluating the limit of the function at that point, we can determine if the function is continuous or not. If the limit exists and is equal to the function's value at that point, then the function is continuous.

3. Can the Limit Definition of Continuity be applied to all functions?

Yes, the Limit Definition of Continuity can be applied to all functions. It is a general definition that can be used to determine the continuity of any type of function, whether it is a polynomial, trigonometric, exponential, or any other type of function.

4. How is the Limit Definition of Continuity different from the traditional definition of continuity?

The traditional definition of continuity states that a function is continuous at a point if the limit of the function exists, the function is defined at that point, and the limit is equal to the function's value at that point. The Limit Definition of Continuity only considers the first two criteria, making it a more general definition that can be applied to a wider range of functions.

5. Can you provide an example of using the Limit Definition of Continuity to prove continuity?

Sure, let's take the function f(x) = x^2 at x = 2. According to the Limit Definition of Continuity, we need to evaluate the limit of f(x) as x approaches 2. Using algebra, we can simplify the limit to 4, which is also the value of f(2). Therefore, the function f(x) = x^2 is continuous at x = 2, as proven by the Limit Definition of Continuity.

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