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Prove using the formal definition of limits: If lim{x->inf} f(x) = -inf and c>0, then lim{x->inf) cf(x) = -inf
You use here the theorem which states that limit of product of functions is product of limits of each function: \[\lim_{x \rightarrow \infty}[f(x)*g(x)]=[\lim_{x \rightarrow \infty}f(x)]*[\lim_{x \rightarrow \infty}g(x)]\] You have here that g(x)=c, which gives: \[\lim_{x \rightarrow \infty}c*f(x)=\lim_{x \rightarrow \infty}c*\lim_{x \rightarrow \infty}f(x)\] Limit of constant is same constant, and limit of function f(x) is given as -inf. You finaly get: \[c*(-\infty)=-\infty\]