Prove lim x → 0 4 − x = 2 using the precise definition of limits.

Question

Prove       lim          x      →      0            4    −    x    =  2 using the precise definition of limits.

Riya Cline · Accepted Answer

First, you can try and prove some important yet simple facts about limits. The following are all equivalent:

\begin{aligned} lim_{x \to a} f (x) = l \\ lim_{h \to 0} f (a + h) = l \\ lim_{h \to 0} f (a - h) = l \\ lim_{x \to a} [f (x) - l] = 0 \end{aligned}

You have

lim_{x \to 0} \sqrt{4 - x} + 2 = 4

or

lim_{h \to 0} \sqrt{4 - h} + 2 = 4

By the above, this is equivalent to

lim_{x \to 4} \sqrt{x} + 2 = 4

So you want to show that for each

ϵ > 0

there is a

δ > 0

such that for all x.

0 < | x - 4 | < δ \Rightarrow | \sqrt{x} - 2 | < ε

But multiplying the conjugate gives

| \sqrt{x} - 2 | = | \frac{x - 4}{\sqrt{x} + 2} | < | x - 4 |

so taking

δ = ϵ

does it.

Prove lim_(x=> infty) sqrt(4-x)=2 using the precise definition of limits.