How Does the (ε, δ)-Definition of Limit Apply to lim x->a x^5=a^5?

[wonnabewith]

I have a question about the (ε, δ)-definition of limit
lim x->a x^5=a^5
I know that |x^5-a^5|<ε
and |x-a|<δ
I was confused when using letters instead of actual number to solve this problem
the goal of this problem is to show that lim x->a x^5=a^5 is true

I will be glad to get some help...
thanks

 

[CompuChip]

If there was not a symbolic constant like a, but a number such as a = 2, would you be able to solve the exercise?

 

[wonnabewith]

the original question is :"use the ε, δ-definition of limit to show that lim x->a x^5=a^5
there is no actual number in it, that's why I'm stuck when I was solving the problem

 

[Some Pig]

For all ε>0, there exists δ=ε/M (M=max|x^4+ax^3+...+a^4|) such that
|x^5-a^5|=|x-a||x^4+ax^3+...+a^4|<δM=ε whenever |x-a|<δ

 

[HallsofIvy]

If there was not a symbolic constant like a, but a number such as a = 2, would you be able to solve the exercise?

the original question is :"use the ε, δ-definition of limit to show that lim x->a x^5=a^5
there is no actual number in it, that's why I'm stuck when I was solving the problem

If you post a problem and one of the most helpful members of this forum asks a question about it, it is a good idea to answer that question!

You said earlier "I was confused when using letters instead of actual number" so compuchip wanted to see how you would do it if it had been "2" instead of "a". Then he could guide you to the "general" idea.

Where it is "a" or "2", the basic idea needed here is that $x^5- a^5= (x- a)(x^4+ ax^3+ a^2x^2+ a^3x+ a^4)$