Updated formal definitions of the limit.

docs/en/mathematics/calculus/limits.md

@@ -44,11 +44,28 @@ If $\lim_{x \to a} f(x) = L$, $\lim_{x \to a} g(x) = M$, and $k$ is a constant t
 The limit $\lim_{x \to a} f(x) = L$ means,
 
 $$
-\forall \varepsilon > 0, \exists \delta \space \mathrm{,s.t.,} \space
+\forall \varepsilon_{> 0} \exists \delta_{>0} \Big[ 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon \Big].
-\forall x \in \mathbb{R}, \space 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon
 $$
 
-For one-sided, infinite and limits at infinity there are similar formal definitions.
+The limit $\lim_{x \to \infty} f(x) = L$ means,
+
+$$
+\forall \varepsilon_{> 0} \exists N_{>0} \Big[x > N \implies |f(x) - L | < \varepsilon \Big].
+$$
+
+The limit $\lim_{x \to a} f(x) = \infty$ means,
+
+$$
+\forall M_{> 0} \exists \delta_{>0} \Big[ 0<|x-a|<\delta \implies f(x) > M \Big].
+$$
+
+The limit $\lim_{x \to \infty} f(x) = \infty$ means,
+
+$$
+\forall M_{> 0} \exists N_{>0} \Big[ x > N \implies f(x) > M \Big].
+$$
+
+For one-sided limits there are similar formal definitions.
 
 ### Example