Limites de um ponto por funçoes continuas

For functions that are continuous at a point, the limit at that point is very easy: it's the same as the function's actual value.

If you get a "nice" function all by itself in a limit that is continuous at the point at which the limit is being taken, you can just evaluate the nice function at the point. As we've rigorously proven, that easy limit is also the only possible limit of the function there.

You only need to be careful when the function isn't defined "nicely" at that point --- when there's a division by zero, for example, or if the function is defined as an infinite sum or product, or something like $\tan (\pi/2)$ is involved. These are just a few of countless examples a function can fail to be "nice". In other cases, limits are very easy.

(Video) Limit Laws to Evaluate a Limit

by PatrickJMT

Here's a video by patrickJMT going over a limit law about the limit of a continuous function at a point.

The law states:

If $f(x)$ is continuous at $a$, then $\lim\limits_{x\rightarrow a}f(x)=f(a)$.

In other words, if a function is continuous at a point and we want to find the limit at that point, all we have to do is plug that point into our function and see what value comes out.

So knowing this, let's start on some problems. The first being to evaluate: $$\lim_{x\rightarrow1}\frac{3x+5}{x+4}$$ The only place this function is not continuous is where it is undefined. Rational functions are undefined when their denominator equals zero. So our function is undefined at $x=-4$ but that's not where we're finding the limit. So we can just plug in $x=1$. $$=\frac{3\color{red}{(1)}+5}{\color{red}{(1)}+4}=\frac{3+5}{1+4}\\ \boxed{=\frac85}$$