Deriving the Limit Definition of |x| Using Symbolic Methods

In summary, the limit definition of |x| is a mathematical expression that represents the limit of a function as the input approaches a specific value. It is derived using symbolic methods and is important in calculus for evaluating the behavior of functions. It can be used to find the limit of other functions and applied in real-world situations to analyze the behavior of systems and quantities.
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Orion1
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Is this the correct symbolic method to differentiate this formula using the limit definition?

[tex]f(x) = x|x|[/tex]

[tex]f'(x) = \lim_{h \rightarrow 0} \frac{(x + h)|(x + h)| - x|x|}{h}[/tex]

[tex]f'(x) = \lim_{h \rightarrow 0} \frac{(x + h)^2 - x^2}{h} = \lim_{h \rightarrow 0} \frac{x^2 + 2hx + h^2 - x^2}{h} = \lim_{h \rightarrow 0} \frac{2hx + h^2}{h} = \lim_{h \rightarrow 0} 2x + h[/tex]
[tex]\boxed{f'(x) = 2|x|}[/tex]

 
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3 cases, x>0, x<0 and x=0.
 

1. What is the limit definition of |x|?

The limit definition of |x| is a mathematical expression that represents the limit of a function as the input approaches a specific value. In the case of |x|, the limit definition is written as lim|x| = |x|, which means that as x approaches a certain value, the output of the function will approach the absolute value of that value.

2. How is the limit definition of |x| derived?

The limit definition of |x| can be derived using symbolic methods, which involve manipulating algebraic expressions to find the limit of a function. In the case of |x|, the limit definition is derived by first rewriting the absolute value function as a piecewise function, and then using the definition of a limit to evaluate the limit as the input approaches the specified value.

3. Why is the limit definition of |x| important in calculus?

The limit definition of |x| is important in calculus because it allows us to evaluate the behavior of a function at a specific point, even if the function is not defined at that point. This is essential for understanding the continuity and differentiability of functions, which are fundamental concepts in calculus.

4. Can the limit definition of |x| be used to find the limit of other functions?

Yes, the limit definition of |x| can be used to find the limit of other functions by using algebraic manipulation and the definition of a limit. However, it may be more efficient to use other methods, such as the laws of limits or L'Hopital's rule, for more complex functions.

5. How is the limit definition of |x| applied in real-world situations?

The limit definition of |x| is applied in real-world situations where the behavior of a function or system needs to be analyzed at a specific point. For example, in physics, the limit definition of |x| can be used to find the velocity or acceleration of an object as it approaches a certain position. In economics, it can be used to calculate marginal cost or marginal revenue as the quantity produced approaches a certain value.

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