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Dethrone
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Prove that $\lim_{{n}\to{\infty}}(3^n+4^n)^{1/n}=4$
Rido12 said:Prove that $\lim_{{n}\to{\infty}}(3^n+4^n)^{1/n}=4$
Rido12 said:Prove that $\lim_{{n}\to{\infty}}(3^n+4^n)^{1/n}=4$
The limit of a function at a point is the value that the function approaches as its input approaches that point. More formally, a limit is a value that a function "gets closer to" as the input of the function gets closer to some other value.
To prove a limit using the formal definition, you must show that for any positive number $\epsilon$, there exists a positive number $N$ such that for all $n>N$, the distance between the limit and the function's output is less than $\epsilon$. This is known as the "epsilon-delta" definition of a limit.
The strategy for proving a limit using the formal definition is to start by assuming that the limit exists and is equal to some value $L$. Then, using algebraic manipulation and inequalities, you must show that for any $\epsilon>0$, you can find a value $N$ such that for all $n>N$, the distance between the limit and the function's output is less than $\epsilon$. If you can find such a value $N$ for any $\epsilon$, then the limit is proven to be equal to $L$.
In this specific limit, we are asked to prove that as $n$ approaches infinity, the expression $(3^n+4^n)^{1/n}$ approaches the value 4. This means that for any $\epsilon>0$, we must find a value $N$ such that for all $n>N$, the distance between $(3^n+4^n)^{1/n}$ and 4 is less than $\epsilon$. By manipulating the expression and setting it equal to $4(3/4)^{1/n}$, we can see that as $n$ gets larger, the term $(3/4)^{1/n}$ approaches 1, making the entire expression approach 4. Therefore, the limit is proven to be 4.
Yes, there are multiple methods for proving limits, such as the squeeze theorem, the limit laws, and using L'Hopital's rule. However, the formal definition is often considered the most rigorous and comprehensive method for proving limits.