Show that f(x) = x^2 + 2x is continuous at 3.
I know that I need to show that if |x - 3| < \delta then |x^2 +2x -15| < \epsilon. But I am not sure what to do after this.
To be precise it has to be proven that for any real number , a real number can be found such that the implication holds.
So the epsilon-delta game begins with epsilon.
Now, for arbitrarily chosen look at the inequality . Treat the expression as you would any other normal variable (think of it as ) and you have a quadratic inequality which holds when .
To put it all in one place you have:
For arbitrarily selected , you can use any and the inequality would imply that the inequality holds. Thus you have proven the continuity at 3.
What Mondreus used as an example is simply a special case of this story: were you to choose then you should use