Calculus

# 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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