My textbook (Calculus Early Transcendentals, 8th edition, by James Stewart ) asks to verify a function is continuous at a point using the definition of continuity and the limit laws. However, why would the text explicitly state to use the limit laws to verify continuity? I'm assuming this implies that direct substitution shouldn't be used and looking at solutions online confirms my suspicions as they all use limit laws as well. However, using direct substitution is much faster than using the limit laws and both methods achieve the same result.
For example, show that $f(x) = (x+2x^3)^4$ is continuous at $x=-1$
By definition of continuity, we're trying to show that $\lim_{x\to-1}(x + 2x^3)^4 =f(-1)$
Show that $f(-1)$ exists
$$f(-1)=(1+2(-1)^3)^4 = 81$$
Now for the limit
Using the limit laws:
$$\lim_{x\to-1}(x + 2x^3)^4$$
$$=[\lim_{x\to-1}x + 2x^3]^4$$
$$=[\lim_{x\to-1}x + \lim_{x\to-1}2x^3]^4$$
$$=[\lim_{x\to-1}x + 2\lim_{x\to-1}x^3]^4$$
$$=[-1 + 2(-1)^3]^4 = 81$$
Using direct substitution:
$$\lim_{x\to-1}(x + 2x^3)^4 = (1+2(-1)^3)^4=81$$
In either method we reach the same result that $\lim_{x\to-1}(x + 2x^3)^4 = 81$ which verifies that $f(x)$ is continuous at $x=-1$ since $\lim_{x\to-1}(x + 2x^3)^4 =f(-1)$
Perhaps I'm missing some important connection between continuity and limits as to why limit laws are used instead of direction substitution to verify continuity?
Best Answer
Maybe it'll help to break down very strictly what we are doing with the limit laws. Firstly,
$${(x + 2x^3)^4 = (x+2x^3)(x+2x^3)(x+2x^3)(x+2x^3)}$$
Now, if we have that the limit
$${\lim_{x\to -1}(x+2x^3)}$$
exists, we can say that
$${\lim_{x\to -1}(x+2x^3)^4 = (\lim_{x\to -1}(x + 2x^3))(\lim_{x\to -1}(x + 2x^3))(\lim_{x\to -1}(x + 2x^3))(\lim_{x\to -1}(x + 2x^3))=(\lim_{x\to -1}(x + 2x^3))^4}$$
this is because the limit of the product is the product of the limits (provided the individual limits exist). This is indeed a limit law.
We know that both ${x}$ and ${2x^3}$ will be continuous functions, and so will their sum. Hence we can say that
$${\lim_{x\to -1}x+2x^3=-1+2(-1)^3=-3}$$
Notice we can only directly substitute ${x=-1}$ because we know ${x+2x^3}$ is continuous. This step becomes invalid if we did not know the continuity of ${x+2x^3}$. And now since the individual limit of ${x+2x^3}$ exists, we have that
$${\lim_{x\to -1}(x+2x^3)^4 = (-3)^4 = 81}$$
And ${f(x)=(x+2x^3)^4}$ also has ${f(-1)=81}$, we can conclude ${f(x)}$ is continuous at ${x=-1}$.
If you just directly plug in the number ${-1}$ into
$${(x+2x^3)^4}$$
and claiming this is
$${\lim_{x\to -1}(x+2x^3)^4}$$
you are assuming that ${(x+2x^3)^4}$ is continuous. So you cannot use this to prove continuity, as this is circular reasoning, as also stated by @JMoravitz
Edit: another way to show continuity is to notice that if ${f(x) = x^4}$ and ${g(x) = x + 2x^3}$, notice that your function is just
$${(x+2x^3)^4 = f(g(x))}$$
Both ${f,g}$ are continuous, and the composition of two continuous functions is continuous - so ${(x+2x^3)^4}$ is continuous