What is the intuitive definition of a limit, and how can it be used to determine if a limit exists?

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∻Solution by Steps∻ ‖ step 1 ⋮ The first limit to evaluate is: $ \lim_{n \to \infty} \sum_{k=1}^{n} \left(3 + \frac{k}{n}\right)^4 \cdot \frac{1}{n} = \frac{781}{5} $ ‖ ‖ step 2 ⋮ The second limit is: $ \lim_{n \to \infty} \sum_{k=1}^{n} \left(3 + \frac{k}{n}\right)^4 \cdot \frac{2}{n} = \frac{1562}{5} $ ‖ ‖ step 3 ⋮ The third limit evaluates to: $ \lim_{n \to \infty} \sum_{k=1}^{n} \left(3 + \frac{22k}{n}\right)^4 \cdot \frac{1}{n} = \frac{443881}{5} $ ‖ ‖ step 4 ⋮ The fourth limit is: $ \lim_{n \to \infty} \sum_{k=1}^{n} \left(3 + \frac{22k}{n}\right)^4 \cdot \frac{2}{n} = \frac{887762}{5} $ ‖ ∻Answer∻ ⚹ The limits evaluated are: $\frac{781}{5}$, $\frac{1562}{5}$, $\frac{443881}{5}$, and $\frac{887762}{5}$ ⚹ ∻Key Concept∻ ⚹ A limit describes the behavior of a function as its input approaches a certain value. ⚹ ∻Explanation∻ ⚹ The limits calculated show how the sums converge to specific values as $ n $ approaches infinity, illustrating the concept of limits in calculus. ⚹◊What is the relationship between the limit of a function as $ x $ approaches $ a $ and the value of the function at $ a $?⍭ Generate me a similar question◊

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