Find the Limit of g(x)sinx as x Approaches 0: Homework Help & Equations

In summary, the given function g(x) is defined as 1 for rational values of x and 0 for irrational values of x. The task is to determine the limit of g(x)sinx as x approaches 0, or prove that it does not exist. After applying the squeeze theorem and evaluating the limit, it is concluded that the limit is 0.
  • #1
klarka
2
0

Homework Statement


g(x) = 1 if x is rational, 0 if x is irrational.

Determine the limit, or prove it doesn't exist.

lim g(x)sinx
x->0

Homework Equations


The Attempt at a Solution


I said the limit is 0.

0 < |g(x)sinx| < |sinx|

lim |g(x)sinx|= 0
x->0
-|sinx| < g(x)sinx < sinx

lim g(x)sinx
x->0

The < are less than or equal. I can't figure out how to put them in correctly.
This was on my final and it's driving me crazy! I couldn't decided if the limit was 0 or didn't exist. If you could point out my mistakes I'd appreciate it!
 
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  • #2
No mistake. The limit is 0. -|sin(x)|<=g(x)*sin(x)<=|sin(x)|. The outer terms go to zero as z->0 so g(x)*sin(x) must also go to zero.
 

FAQ: Find the Limit of g(x)sinx as x Approaches 0: Homework Help & Equations

What is the definition of a limit?

A limit is the value that a function approaches as the input variable gets closer and closer to a particular value. In other words, it is the value that a function is "heading towards" as the input value approaches a certain value.

How do I find the limit of a function?

To find the limit of a function, you can either use algebraic techniques, such as factoring and simplifying, or you can use graphical techniques, such as graphing the function or using a table of values. In this case, to find the limit of g(x)sinx as x approaches 0, you would plug in increasingly smaller values for x and observe the corresponding outputs.

What is the significance of finding a limit?

Finding a limit can help us understand the behavior of a function at a particular point, even if the function is undefined at that point. It can also help us determine the continuity of a function, which is important in many mathematical and scientific applications.

What does it mean if a limit does not exist?

If a limit does not exist, it means that the function does not approach a single value as the input variable approaches a certain value. This could be due to a jump or discontinuity in the function, or the function may oscillate or become unbounded as the input value gets closer to the specified value.

Can the limit of a function be different at different points?

Yes, the limit of a function can be different at different points. This is because the behavior of a function can vary depending on the input value and how close it is to the specified point. It is important to consider the limit at each point separately to fully understand the behavior of the function.

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