Proof of power rule of limit laws

In summary, the Power Rule states that if r and s are integers with no common factor and s is not equal to 0, then the limit of f(x) raised to the power of r/s as x approaches c is equal to the limit of f(x) raised to the power of r/s as x approaches c, provided that the limit of f(x) raised to the power of r/s is a real number. The proof is related to the Sandwich Theorem and can be shown using induction. To find the limit of f^(1/m)(x), one can define g(x) as f^(1/n)(x) and take the limit of g^n(x) as x approaches c.
  • #1
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Homework Statement


Power Rule: If r and s are integers with no common factor and s=/=0, then
lim(f(x))r/s = Lr/s
x[tex]\rightarrow[/tex]c
provided that Lr/s is a real number. (If s is even, we assume that L>0)
How can I prove it?

Homework Equations





The Attempt at a Solution


I heard that the proof is related to The Sandwich Theorem
 
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  • #2
From [itex]\lim_{x\rightarrow c} f(x)g(x)= \left(\lim_{x \rightarrow c}f(x)\right)\left(\lim_{x \rightarrow c}g(x)[/itex] you should be able to prove that [itex]\lim_{x\rightarrow c}f^2(x)= \left(\lim_{x\rightarrow c} f(x)\right)^2[/itex]. Then use induction to prove that [itex]\lim_{x\rightarrow c}f^n(x)= \lim_{x\rightarrow c}\left(f(x)\right)^n[/itex]. That's the easy part.

For [itex]\lim_{x\rightarrow c} f^{1/m}(x)[/itex], assuming that limit exists, define [itex]g(x)= f^{1/n}(x)[/itex] and look at [itex]lim_{x\rightarrow c}g^n(x)[/itex].
 
  • #3
Thank you for answering me. But how can we assume that lim f^1/m(x) exists?
x->c
 

What is the power rule of limit laws?

The power rule of limit laws is a rule that allows us to simplify limits involving powers of a variable. It states that for a function f(x) = x^n, where n is any real number, the limit as x approaches a of f(x) is equal to a^n, where a is a real number.

How is the power rule of limit laws used in calculus?

The power rule of limit laws is used in calculus to help us evaluate limits involving powers of a variable. It is particularly useful in finding the derivatives of polynomial functions.

What are the other limit laws in calculus?

Some other limit laws in calculus include the sum and difference rules, the product rule, the quotient rule, and the chain rule. These rules help us to evaluate limits of more complex functions by breaking them down into simpler parts.

Can the power rule of limit laws be extended to other types of functions?

Yes, the power rule of limit laws can be extended to other types of functions, such as trigonometric, exponential, and logarithmic functions. However, the power rule may need to be modified for these functions depending on their properties.

How can the power rule of limit laws be applied in real-world situations?

The power rule of limit laws has many real-world applications, such as in physics, engineering, and economics. It can be used to model and analyze various phenomena, such as population growth, radioactive decay, and electrical circuits.

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