Limit of Real Function/Examples/Root of x at 1

Example of Limit of Real Function

 * $\ds \lim_{x \mathop \to 1} \sqrt x = 1$

Proof
By definition of the limit of a real function:
 * $\ds \lim_{x \mathop \to a} \map f x = A$


 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - a} < \delta \implies \size {\map f x - A} < \epsilon$
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - a} < \delta \implies \size {\map f x - A} < \epsilon$

In this instance, we have:
 * $\map f x = \sqrt x$
 * $A = 1$

So:

for $0 < x < 2$.

Let $\delta = \epsilon$.

Let $0 < \size {x - 1} < \delta = \epsilon$.

Then we obtain: