Is f(x) continuous at x=0?

In summary, by using the squeeze theorem, it can be shown that the function f(x) is continuous at x=0. This is because the limit of the function approaches 0 as x approaches 0, satisfying the definition of continuity at a point.
  • #1
kidia
66
0
Using the inequalities -1 = < sin 1/x = < 1 for x not equal to 0 determine wether or not the function
f(x)= {sinxsin1/x x is not equal to 0
f(x)= {0 x=0

continuous at x=0
 
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  • #2
Hmm...

Well, we obviously must find whether or not
[tex]\lim_{x\rightarrow0}\sin{x}\sin{\frac{1}{x}}=0[/tex]

Which is a bit of a challenge since [tex]\lim_{x\rightarrow\infty}\sin{\frac{1}{x}}[/tex] is undefined.
Looks like squeeze theorem time.
For all x, [tex]-|x|\leq\sin{x}\leq|x|[/tex] and [tex]|\sin{\frac{1}{x}}|\leq1[/tex].
Therefore, [tex]-|x|\leq\sin{x}\sin{\frac{1}{x}}\leq|x|[/tex] for all x.
Since [tex]\lim_{x\rightarrow0}-|x|=\lim_{x\rightarrow0}|x|=0, \lim_{x\rightarrow0}\sin{x}\sin{\frac{1}{x}}=0[/tex]

And thus it is continuous
 
  • #3


The function f(x) is continuous at x=0 because it satisfies the definition of continuity, which states that a function is continuous at a point if the limit of the function at that point is equal to the value of the function at that point.

In this case, as x approaches 0, the limit of sin 1/x is equal to 0, and the value of f(x) at x=0 is also 0. This means that the function is continuous at x=0.

Furthermore, the given inequalities -1 ≤ sin 1/x ≤ 1 for x ≠ 0 also support the continuity of the function at x=0. These inequalities show that the values of sin 1/x are always bounded between -1 and 1, which means that the function does not have any sudden jumps or breaks at x=0.

Therefore, we can conclude that the function f(x) is continuous at x=0.
 

What is a continuous function?

A continuous function is a function in mathematics that has no sudden changes or breaks in its graph. This means that the values of the function change gradually as the input values change, without any abrupt jumps or interruptions.

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

A continuous function is one that is unbroken and smooth, while a discontinuous function has breaks or gaps in its graph. This means that the value of a discontinuous function can suddenly change, while the value of a continuous function changes gradually.

How can you determine if a function is continuous?

A function is continuous if its graph is a single unbroken curve. This means that there are no sudden changes or gaps in the graph and that the value of the function changes gradually as the input values change. To determine if a function is continuous, you can graph it or use the formal definition of continuity.

What are some real-life examples of continuous functions?

Some examples of continuous functions in real life include temperature over time, distance traveled over time, and population growth over time. These functions change gradually and do not have sudden changes or breaks in their values.

Why are continuous functions important in mathematics and science?

Continuous functions are important in mathematics and science because they represent real-world phenomena that change gradually over time or space. They allow us to model and solve complex problems and make predictions about the behavior of systems. Continuous functions also have many useful properties that make them easier to work with compared to discontinuous functions.

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