Answer
$$\eqalign{
& {\text{The limit of }}f\left( x \right){\text{as }}x{\text{ approaches }}a{\text{ is }}L,{\text{ written}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {\lim }\limits_{x \to a} f\left( x \right) = L \cr
& {\text{if for }}any\,\varepsilon > 0{\text{ there is a corresponding number }}\delta {\text{ > 0 such that}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| {f\left( x \right) - L} \right| < \varepsilon {\text{ whenever }}0 < \left| {x - a} \right| < \delta \cr} $$
Work Step by Step
$$\eqalign{
& {\text{The precise definition of }}\mathop {\lim }\limits_{x \to a} f\left( x \right) = L{\text{ is:}} \cr
& {\text{The limit of }}f\left( x \right){\text{as }}x{\text{ approaches }}a{\text{ is }}L,{\text{ written}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathop {\lim }\limits_{x \to a} f\left( x \right) = L \cr
& {\text{if for }}any\,\varepsilon > 0{\text{ there is a corresponding number }}\delta {\text{ > 0 such that}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| {f\left( x \right) - L} \right| < \varepsilon {\text{ whenever }}0 < \left| {x - a} \right| < \delta \cr} $$
