Is |\sin x|=-\sin x as x Approaches 0-?

In summary, when approaching 0 from the left, the absolute value of sinx becomes -sinx. It is necessary to consider values of x between -pi and 0 in order to remove the absolute value sign and find the limit of 0.
  • #1
Kamataat
137
0
Is it correct that [itex]|\sin x|=-\sin x[/itex] as [itex]x\rightarrow 0-[/itex]?

It's clear for a function like [itex]y=x[/itex], but I ask the question since the sine oscillates from positive to negative values, so for different x's the abs. value is either pos. or neg. Or do I only need to consider values of x that are between -pi and 0?

I know the limit is zero, I just need to know how to write the intermediate step of getting rid of the abs. value sign.

- Kamataat
 
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  • #2
If you're approaching 0 from the left (so bottom, smaller values i.e. negatives) then the sine is negative there, so |sinx| indeed becomes -sinx.
 
  • #3
Thanks for the quick reply!

- Kamataat
 

FAQ: Is |\sin x|=-\sin x as x Approaches 0-?

What is the limit of absolute value?

The limit of absolute value is the value that a function approaches as its input approaches a specific value. It is denoted by the notation "lim|x|".

How do you find the limit of absolute value?

To find the limit of absolute value, you can evaluate the function at the given input value and see what value it approaches as the input gets closer to that value. You can also use algebraic manipulation or graphing to determine the limit.

What is the difference between left and right limits of absolute value?

The left limit of absolute value is the value that the function approaches from the left side of the input value, while the right limit is the value that the function approaches from the right side. In some cases, the left and right limits may be different, indicating that the overall limit does not exist.

Why is the limit of absolute value important in calculus?

The limit of absolute value is important in calculus because it allows us to understand the behavior of a function at a specific point. It helps us determine if a function is continuous at a certain point, and is a fundamental concept in the study of derivatives and integrals.

What are some real-life applications of the limit of absolute value?

The limit of absolute value has many real-life applications in fields such as physics, engineering, and economics. For example, it can be used to calculate the velocity of an object at a specific time, determine the maximum load a bridge can withstand, or analyze the demand and supply of a product in the market.

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