Question

Prove using the formal definition of limits:
If lim{x->inf} f(x) = -inf and c>0, then lim{x->inf) cf(x) = -inf

Answer 1

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: \[c*(-\infty)=-\infty\]

Answer 2

Ty! :)