Continuity of quotient map at $\infty/\infty$

Question

I have a mapping of the form $H(x) := f(x) / g(x), x \in [0,1]$, with $f$ and $g$ continuous on $[0,1)$ and both infinite on $1$, however $\lim_{x\rightarrow 1} |f(x)/g(x) a = :| < \infty$. I can therefore define $H(1) := a$. Does this ensure that $H$ is continuous in $1$? Or do I need additional assumptions?

Answer 1

In general, if $f:[a,b)\longrightarrow \mathbb{R}$ is continuous, and $\underset{x\longrightarrow b}{lim}f$ exists, you can define $\tilde{f}(b)=\underset{x\longrightarrow b}{lim}f$, and you will receive a continuous function $\tilde{f}:[a,b]\longrightarrow \mathbb{R}$.