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Limits of Sequences and Series

Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".

Properties of Limits

If sequence \(\{a_n\}\) satisfies \[\displaystyle \lim_{n \to \infty}(2n-1)a_n=16,\] what is the value of \(\displaystyle \lim_{n \to \infty}n a_n\)?

What is the value of \[\lim_{x \to 0} \frac{5}{x}\left(\frac{1}{x+1}-\frac{1}{5x+1}\right)?\]

Given that \( \displaystyle \lim_{n \to \infty} \left\{ a_n \right\} = 10 \) , and \( \displaystyle \lim_{n \to \infty} \left\{ b_n \right\} = 11 \), and \( \left\{ c_n \right\} = \left\{ a_n + b_n \right\} \), evaluate: \[ \displaystyle \lim_{n \to \infty} \left\{ c_n \right\}. \]

Below is the graph of \(y=f(x),\) with \(a=5\), \(b=2\) and \(c=9\). What is the value of \[\lim_{x \to a^+} f(x)+\lim_{x \to a^-} f(x)?\]

Below is the graph of function \(f(x).\) If \[a=10, b=20, f(a)=c=10,\] what is the value of \(\displaystyle \lim_{x \to a} f(x)?\)

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