Is the Function f=tan(2x)/x Continuous at x=0?

In summary, a continuous function is a mathematical concept where there are no sudden jumps or breaks in the graph of the function, and it is determined by three conditions: the function is defined at every point in its domain, the limit of the function exists at every point in its domain, and the limit of the function at a specific point is equal to the value of the function at that point. This concept is important in mathematics as it allows for more precise analysis and predictions in various fields, such as physics, engineering, and economics. It is also commonly used in real-world applications to model natural phenomena and optimize processes.
  • #1
cbarker1
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Let $f=tan(2x)/x$, x is not equal to 0.

Can the f be defined at x=0 such that it is continuous? I answered yes. I am wondering if the answer is correct. Thank you for your help

CBarker1
 
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  • #2
Compute limit at $0$. What do you get?
 
  • #3
I got 2.
 
  • #4
Cbarker1 said:
I got 2.

So,
$$\lim_{x\to 0} \frac{\tan 2x}{x} = 2$$
Now define the function,
$$ f(x) = \left\{ \begin{array}{ccc}(\tan x)/x & \text{if} & x\not = 0 \\ 2 & \text{if}& x=0 \end{array} \right. $$

This function is continuous everywhere because at $0$ we have $\lim_{x\to 0}f(x) = f(0) = 2$.
 

FAQ: Is the Function f=tan(2x)/x Continuous at x=0?

What is a continuous function?

A continuous function is a mathematical concept where there are no sudden jumps or breaks in the graph of the function. This means that as the input values of the function change, the output values also change smoothly without any abrupt changes.

How can you tell if a function is continuous?

A function is considered continuous if it satisfies three conditions: 1) the function is defined at every point in its domain, 2) the limit of the function exists at every point in its domain, and 3) the limit of the function at a specific point is equal to the value of the function at that point.

What is the difference between a continuous function and a discontinuous function?

A continuous function has no breaks or abrupt changes in its graph, while a discontinuous function has at least one point where there is a sudden jump or break in the graph. This means that a discontinuous function does not satisfy the three conditions of a continuous function.

Why is the concept of continuity important in mathematics?

The concept of continuity is important in mathematics because it allows us to study and analyze functions in a more precise and accurate way. It also helps us to make predictions and solve problems in various fields such as physics, engineering, and economics.

How can continuity be used in real-world applications?

Continuity is used in various real-world applications, such as predicting the behavior of a system over time, optimizing processes, and modeling natural phenomena. For example, in physics, continuity is used to describe the motion of objects, while in economics, it is used to model the behavior of markets.

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