Let's say we have $f(x),g(x)$ that are defined in the area of $x_0$ (not including $x_0$)
And for every $x$ in the area of $x_0$ exist: $f(x) \ge g(x)$ and $\lim_{x\to x_0}g(x) = \infty$
How do I prove using the definition of the limit that $\lim_{x\to x_0}f(x) = \infty$
Best Answer
Formally, a limit like this means that for all $N > 0$, there exists some $\delta > 0$ such that $0 < |x - x_0| < \delta \implies g(x) > N$. Now you can utilize the fact that if $a \geq b$ and $b > c$, then $a > c$ to show that this also applies to $f$.