Suppose that $\lim_{x\to a}f(x)=+\infty$ and $g(x)$ is bounded. Show that $\lim_{x\to a}(f(x)-g(x))=+\infty$ by using definition
Here is what I think. Please!
Since $g(x)$ is bounded. Then $|g(x)|\leq M$
And $\lim_{x\to a}f(x)=+\infty$ We have $A >0,\exists \delta$ such that $|f(x)|>A$, where $|x-a|<\delta$
You can prove it like this also:
Since $f(x)\to \infty$ as $x\to a$, it follows that for any $A\gt 0$ there exists $h\gt 0$ such that $0\lt |x-a|\lt h\implies f(x)\ge A$.
Since $g$ is bounded, it follows that there exists some $M\gt 0$ such that $|g(x)|\lt M$ for all $x$. It follows that $ -g(x)\gt -M$
Choosing $A=A'+M$, where $A'\gt 0$ is any arbitrary positive no. Clearly there exists $h'$ such that $0\lt |x-a|\lt h'\implies f(x)\ge A'+M$
We have: $f(x)-g(x)\gt A'$ whenever $0\lt |x-a|\lt h'$. Since $A'\gt 0$, is arbitrary it follows that "For every $A'\gt 0$, there exists $h'\gt 0$ such that $0\lt |x-a|\lt h'\implies f(x)-g(x)\ge A'$", which proves that $f(x)-g(x)\to \infty$ as $x\to a$