For the given continuous function, value of *c*, and ε, find *f*(*c*) and an appropriate δ as guaranteed by the continuity of *f*.

*x*^{2} + 2 , *c* = -2 , δ = 0.2

Hint : Squares have both positive and negative square roots.

Answer

Solve the inequality for *x*.

Add 2 to both sides to find

14 < 5*x* < 16,

then divide everything by 5 to find

Evaluate the fractions to find

2.8 < *x* < 3.2

We write down the inequality

*f*(c) - ε < *f*(*x*) < *f*(c) + ε

and fill in what we were given for *c*, *f*, and ε:

1 - 0.1< *f*(*x*) < 1 + 0.1

or

0.9 < e^{x} < 1.1.

Solve the inequality for *x*. Take ln of each part of the inequality to find

ln(0.9) < *x* < ln(1.1).

Evaluating and rounding the natural logs,

-1.05 < *x *< 0.095.

Subtract *c* from all parts of the inequality and find δ. In this case *c* = 0, therefore the inequality stays the same. Since the bound on the right-hand side is smaller, we use that one for δ to find

δ = 0.095.

Since *c* = -2,

*f*(c) = *f*(-2) = (-2) ^{2} + 2 = 6.

We're given ε = 0.2.

We write down the inequality

*f*(c) - ε < *f*(*x*) < *f*(c) + ε

and fill in what we were given for *c*, *f*, and ε:

6 - 0.2 < *f*(*x*) < 6 + 0.2 or 5.8 < *x*^{2} + 2 < 6.2.

Solve the inequality for *x*. Subtract 2 from all parts of the inequality to find

3.8 < *x*^{2} < 4.2, then take square roots and round to find 1.995 < *x* < 2.005.

**Notice this doesn't make sense.** *x* is supposed to be close to *-2*, not close to 2. The problem is that we need to take negative square roots, which gives us

-1.995 >* x* > -2.005

(also notice that the direction of the inequalities need to switch, since -1.995 is *bigger* than -2.005). Subtract *c* from all parts of the inequality and find δ. In this case *c* = -2, we find

-1.995 - (-2) > *x *- (-2) > -2.005 - (-2)

0.005 > *x *- (-2) > -0.005.

Therefore *x* need to be within 0.005 of *-2*, therefore

δ = 0.005.