$C.\lim_{x \to \infty}f(x)=\lim_{x \to \infty}(C.f(x))$ but $0.\lim_{x \to \infty}f(x) $is consider indeterminate form while $\lim_{x \to \infty}(0.f(x))=0 $? so why using L'Hospital' when you can simple insert zero into the limit? thanks.
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There is nothing indeterminate about $0\cdot \lim_{x\to\infty} f(x)$ if you assume that the limit exists (i.e. exists as a real number; note that limits that "are $\pm\infty$" are improper and not really limits in the strict sense).
Similarly, the rule
$$C\cdot \lim_{x\to\infty} f(x) = \lim_{x\to\infty} C\cdot f(x)
$$
is only valid if the limits exist (again as real numbers).
