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When does a sequence get within $d$ of its limit?
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

$x_n=\frac{an^2+b}{cn^2+d}$ . Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| < 10 ^{-r},\;n\geq N$ , $2\leq r \leq 6$ . Determine whether the sequence is increasing, decreasing or neither.
Find point when a sequence becomes strictly decreasing
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$x_n=n^k t^n$ where $k$ is a positive integer and $t$ a real number with $0 < t<1$ . Find the smallest integer $N$ such that $(m+1)^k t^{m+1} \leq m^k t^m$ for all $m \geq N$ .
Find limit of a sequence
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Let $x_n=\frac{an+b}{cn+d},\;\;n=1,\;2\ldots$ . Find $\lim_{x \to\infty} x_n=L$ and find least $N$ such that $|x_n-L| \le 10^{-r},\;n \geq N,\;r \in \{2,\;3,\;4\}$ .
When is a sequence within d of its limit?
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Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

$x_n=\frac{an+b}{cn+d}$ . Find the least integer $N$ such that $\left|x_n -\frac{a}{c}\right| \le 10 ^{-r},\;n\geq N$ , $2\leq r \leq 6$ .