Question

Use the definition of continuity and the properties of the limits to show that the function is continous at the given number "a". f(x)= (x+2x^3)^4, "a"=-1

I know that I have to show that the limits approaching from the left and right are both the same(lim f(x)-> a+ and lim f(x)->a-) but I don't know how to show/do that.

Answer 1

first off it is clear that any polynomial is continuous everywhere, so you can just replace \(x\) by \(-1\) but i think i can tell you what they want you to write

Answer 2

they want
\[\lim_{x\to a} (x+2x^3)^4=\left(\lim_{x\to a}(x+2x^2)\right)^4\] i.e. you can bring the limit inside the power because
\[\lim(f^n(x))=(\lim f(x))^n\]
then
\[\left(\lim_{x\to a}x+2\lim_{x\to a}x^2\right)^4\]

Answer 3

then use one more time to take the limit inside the power in the second term and get
\[\left(\lim_{x\to a}x+2(\lim_{x\to a}x)^2\right)^4\]

Answer 4

and by sheer obviousness \(\lim_{x\to a}x=a\) so you get
\[\left(a+2a^2\right)^4=f(a)\]

Answer 5

oh i also used that \(\lim_{x\to a}cf(x)=c\lim_{x\to a}f(x)\) to pull that two out front of the second term inside the parentheses

Answer 6

you can replace \(a\) by \(-1\) but then again it works perfectly well for any number \(a\)

Answer 7

so it's just the limit laws? I thought it was about continuity