For the following exercises, suppose that lim x -> a f(x) = L and lim x -> a g(x) = M both exist. Use the precise definition of limits to prove the following limit laws: lim x -> a (f(x) + g(x)) = L + M
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The limit of a function f(x) as x approaches a value c is defined as follows: lim x->c f(x) = L if for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. Show more…
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$$\begin{aligned} &\text { Proof of Limit Law 2 Suppose } \lim f(x)=L \text { and } \lim _{x \rightarrow a} g(x)=M\\ &\text { Prove that } \lim _{x \rightarrow a}[f(x)-g(x)]=L^{\infty}-M \end{aligned}$$
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For the following exercises, suppose that $$\lim _{x \rightarrow a} f(x)=L$$ and $$\lim _{x \rightarrow a} g(x)=M$$ both exist. Use the precise definition of limits to prove the following limit laws: $$\lim _{x \rightarrow a}[c f(x)]=c L$$ for any real constant $c$ (Hint: Consider two cases: $c=0$ and $c \neq 0 .$)
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