How do you show the function is continuous at the given number a #f(x)=(x+2x^3)^4#, a=-1?
The limit exists, so now you have to calculate
The calculation shows, that
Okay. Maybe I should do an epsilon/delta proof here, though it's not pleasant. It can be done, though you might not want to do it unless your life depended on it (lol).
First, a bit of scratchwork. The function
The distance between
Without loss of generality, we can assume that
Somebody please let me know if I made any mistakes.
Now here's an outline of the proof. Let
From this point, do the same kind of work as shown above (fill in the gaps
That finishes the proof. It's nasty problems like this that give abstract theorems their value (like the abstract fact that any polynomial is continuous for all
"How you show" depends on what tools you have to work with.
I assmue that you know that
Have you proven that all polynomials are continuous on their domains ?
If so, then point out that
If you do not have that tool to work with, have you got:
The limit of a power is the power of the limit AND the limit of a constant multiple is the constant multiple times the limit AND the limit of a sum is the sum of the limit. (These often appear as "Limit Laws" or "Properties of Limits")?
# = [lim_(xrarr-1)(x)+ lim_(xrarr-1)(2x^3)]^4# #color(white)"sss"#(sum)
# = [lim_(xrarr-1)(x)+ 2lim_(xrarr-1)(x^3)]^4# #color(white)"sss"#(constant multiple)
# = [lim_(xrarr-1)(x)+ 2(lim_(xrarr-1)(x))^3]^4# #color(white)"sss"#(power)
#= [(-1)+2(-1)^3]^4# #color(white)"ssssss"#( #lim_(xrarra) x = a#)
The last line above is clearly equal to:
So, by the definition of "continuous at
If you have none of these , you'll need an answer like Bill Kinney's answer.