ModalNekat said:Homework Statement
Limit x approached zero from right which is X * [|1/X|]
Homework Equations
Since i Don't know how to draw the graphs, then i don't have an equations
The Attempt at a Solution
I've tried to drew the graphic, but from the graphs, I've concluded that lim x aprroached zero from right [|1/x|] does not exist because 1/0 = undefined.
Actually, the answer is 1, but still i don't know how to solve this.
So, guys, please help me up!Thanks
Whoops, I'm sorry for misunderstanding, looks like [| |] is not dealing with Greatest INteger Function. Then, what i mean with [|x|] is [[ x ]], sorry..VietDao29 said:Is this your limit:
[tex]\lim_{x \rightarrow 0 ^ {+}} \left( x \left| \frac{1}{x} \right| \right)[/tex]?
Well, the first thing when dealing with absolute value, is to take out all absolute signs.
You know that:
[tex]|a| = \left\{ \begin{array}{ll} a & \mbox{, if } a \geq 0 \\ -a & \mbox{, if } a < 0 \end{array} \right.[/tex]
So, when x tends to 0+, (i.e it tends to 0 from the right), is 1/x positive or negative? Can you break absolute signs?
After breaking (taking out) all the absolute signs, you should be arriving to the final answer shortly. :)
ModalNekat said:Whoops, I'm sorry for misunderstanding, looks like [| |] is not dealing with Greatest INteger Function. Then, what i mean with [|x|] is [[ x ]], sorry..
Yeah, i mean it's the ceiling function. Thanks a lot, right now it could be derived.Actually i got this problems from Purcell's Calculus Book, do u have one on you??VietDao29 said:You mean the http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" , right? Or are you referring to the Floor Function?
[tex]\lim_{x \rightarrow 0 ^ {+}} \left( x \left\lceil \frac{1}{x} \right\rceil \right)[/tex]
If you really mean the Ceiling Function, then from its definition, we can derive to the following inequality:
[tex]\frac{1}{x} \leq \left\lceil \frac{1}{x} \right\rceil \leq \frac{1}{x} + 1[/tex]
Can you find the limit of the expression based on this inequality?
emyt said:what is the answer?
emyt said:the interval between 1/x and 1/x + 1 is fairly large though?
A limit in mathematics is a fundamental concept that describes the behavior of a function as the input approaches a certain value. It is used to determine the value that a function approaches as the input gets closer and closer to a particular point.
Limits are essential in mathematics because they help us understand the behavior of functions and their graphs. They are also used to define important concepts such as derivatives and integrals, which are crucial in calculus and other branches of mathematics.
The process of solving limit problems involves evaluating the function at different points, both numerically and graphically, and observing the behavior of the function as the input approaches the desired value. Algebraic techniques, such as factoring and simplifying, are also useful in solving limit problems.
The most common types of limit problems include evaluating limits at infinity, evaluating limits involving trigonometric functions, and evaluating limits involving rational functions. Other types may involve piecewise-defined functions, exponential functions, or logarithmic functions.
To check if your limit solution is correct, you can use a graphing calculator or an online limit calculator to compare your answer with the expected result. You can also verify your solution algebraically by plugging in the value of the limit into the original function and simplifying it to see if it matches your answer.