- #1
asdf60
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how would i prove that lim of x^4 as x->p is p^4? x^4-p^4 = (x-p)(x+p)(x^2+p^2). I'm having trouble controlling the (x+p)(x^2+p^2) term without having to resort to proving for p > 0 and p < 0 seperately.
When we say that the limit of x^4 as x approaches p is p^4, we mean that as x gets closer and closer to the value of p, the value of the expression x^4 also gets closer and closer to the value of p^4. In other words, the limit is the value that the function approaches as x gets infinitely close to p.
To prove this limit, we need to show that as x gets arbitrarily close to p, the difference between the value of x^4 and p^4 approaches 0. This can be done by using the formal definition of a limit, which involves using epsilon-delta notation and showing that for any positive value of epsilon, we can find a corresponding value of delta that satisfies the definition.
Proving this limit is important because it helps us understand the behavior of the function x^4 as x gets closer to p. It also allows us to make accurate predictions and calculations involving this function, which can have real-world applications in fields such as physics, engineering, and economics.
Some common techniques used to prove limits include algebraic manipulation, substitution, and the use of limit laws such as the sum, difference, product, and quotient rules. In more advanced cases, we may also use L'Hôpital's rule, squeeze theorem, or Taylor series to prove limits.
No, a graph alone is not enough to prove the limit of x^4 as x approaches p. While a graph can help us visualize the behavior of the function, it cannot provide a formal proof. To prove a limit, we need to use mathematical techniques and definitions, as well as logical reasoning.