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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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