# How do you show the function is continuous at the given number a #f(x)=(x+2x^3)^4#, a=-1?

##### 3 Answers

The function

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

**First attempt**

Have you proven that all **polynomials are continuous on their domains** ?

If so, then point out that

**Second attempt**

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")?

If so:

# = [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# )

#=81#

The last line above is clearly equal to:

That is:

So, by the definition of "continuous at

**Third**

If you have **none of these** , you'll need an answer like Bill Kinney's answer.