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Question

# Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $\lim _{x \rightarrow 2}\left(\frac{x}{x+1}+\frac{3}{x-1}\right)=\lim _{x \rightarrow 2} \frac{x}{x+1}+\lim _{x \rightarrow 2} \frac{3}{x-1}$

Solution

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Let's solve given problem.

$\lim_{x\to2}\left(\frac{x}{x+1}+\frac{3}{x-1}\right)$

Using property

If both limits exist then

$\lim_{x\to a}(f(x)+g(x))=\lim_{x\to a}f(x)+\lim_{x\to a}g(x)$

we have

$\lim_{x\to2}\frac{x}{x+1}=\frac{2}{2+1}=\frac{2}{3}$

$\lim_{x\to2}\frac{3}{x-1}=\frac{3}{2-1}=3$

so

$\lim_{x\to2}\left(\frac{x}{x+1}+\frac{3}{x-1}\right)=\lim_{x\to2}\frac{x}{x+1}+\lim_{x\to2}\frac{3}{x-1}$

Therefore, the given statement is $\color{#19804f}{\text{true}}$.

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