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## Homework Statement

Show, from the definition of what it means for a function to converge to a limit, that the sequence ##\left\{x^t\right\}_{t=1}^{\infty}## with ##x^t = \frac{2t+5}{t^2+7}## converges to ##0## as ##t## goes to infinity.

## Homework Equations

A sequence converges to ##x^0 \in X## if for any ##\epsilon > 0##, there is ##N \in \mathbb{N}## such that if ##t > N##, then ##d(x^t,x^0) < \epsilon##.

## The Attempt at a Solution

To show that ##x^t = \frac{2t+5}{t^2+7}## converges to ##0## we must, for any ##\epsilon > 0##, find a value ##N## such that if ##t > N##, then

$$\left|\frac{2t+5}{t^2+7} - 0\right| = \left|\frac{2t+5}{t^2+7}\right| < \epsilon$$.

Now sure how to simplify ##x^t## to show that it is less than or equal to some much simpler expression in ##t## that can clearly be made less than any given ##\epsilon## by choosing ##n## large enough. Please help.