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I am trying to find a more consistent way to find and prove that a limit exists in Real Analysis with concrete numbers.

I am familiar with the basic process of setting

$\lvert x_n - a \rvert < sigma$ and then finding an N such that this is always true for n>=N, but I am struggling to apply it to actual examples. For example, what would I actually do in this problem to prove that the limit is 1/2:

$\sqrt{x^2 +x} - x $

Similarly, what would I do in this problem to prove that the limit is 3:

$(3^n + 2^n)^{\frac 1 n}$

For the first one, I plugged in my sequence and my 1/2 as the limit that we know, and ended up with $1/N > 1/(sigma + 1/2)$

but that doesn't seem right, and doesn't seem like how these problems are normally solved. Any guidance would be appreciated

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I think that you mean: $\sqrt{n^2 +n} - n^2 \to 1/2$ as $n \to \infty$

Look at $\sqrt{n^2 +n} - n^2= \frac{(\sqrt{n^2 +n} - n^2)(\sqrt{n^2 +n} + n^2)}{\sqrt{n^2 +n} + n^2}=\frac{n}{\sqrt{n^2 +n} + n^2}$.

Can you take it from here ?

To your 2. sequence:

$3=(3^n)^{1/n} \le (3^n+2^n)^{1/n} \le (2*3^n)^{1/n}=2^{1/n}*3$.

Now squeeze !

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