# Using the definition of convergence, how do you prove that the sequence # lim (3n+1)/(2n+5)=3/2# converges?

I assume meant limit at infinity, that is, to show that

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To prove that the sequence ( \lim_{n \to \infty} \frac{3n+1}{2n+5} = \frac{3}{2} ) converges using the definition of convergence, we need to show that for any positive real number ( \epsilon ), there exists a positive integer ( N ) such that for all ( n \geq N ), the absolute difference between ( \frac{3n+1}{2n+5} ) and ( \frac{3}{2} ) is less than ( \epsilon ).

Let ( \epsilon > 0 ) be given. We aim to find ( N ) such that for all ( n \geq N ), ( \left| \frac{3n+1}{2n+5} - \frac{3}{2} \right| < \epsilon ).

We start by simplifying the expression: [ \left| \frac{3n+1}{2n+5} - \frac{3}{2} \right| = \left| \frac{6n+2 - 3(2n+5)}{2(2n+5)} \right| = \left| \frac{6n+2 - 6n - 15}{4n+10} \right| = \left| \frac{-13}{4n+10} \right| = \frac{13}{4n+10} ]

To ensure that ( \frac{13}{4n+10} < \epsilon ), we can set: [ \frac{13}{4n+10} < \epsilon ] [ 13 < \epsilon(4n+10) ] [ \frac{13}{\epsilon} < 4n + 10 ] [ \frac{13}{4\epsilon} - \frac{10}{4} < n ] [ \frac{13 - 10\epsilon}{4\epsilon} < n ]

Now, let ( N = \lceil \frac{13 - 10\epsilon}{4\epsilon} \rceil + 1 ), where ( \lceil x \rceil ) denotes the smallest integer greater than or equal to ( x ). Then, for all ( n \geq N ), we have: [ n \geq \frac{13 - 10\epsilon}{4\epsilon} + 1 ] [ n > \frac{13 - 10\epsilon}{4\epsilon} ] [ \frac{13}{4\epsilon} - \frac{10}{4} < n ]

Therefore, for all ( n \geq N ), ( \frac{13}{4n+10} < \epsilon ), which completes the proof that the sequence converges to ( \frac{3}{2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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