Find the following limits or state that they do not exist. Assume a, b, c,...

Concept:
The calculus helps in understanding the changes between values that are related by a function. It is used in different areas such as physics, biology and sociology and so on. The calculus is the language of engineers, scientists and economics.
Given:
${\displaystyle \underset{x\to 3}{lim}\frac{\frac{1}{x^2+2x}-\frac{1}{15}}{x-3}}$
Solution:
Thus the given function is as follows,
${\displaystyle \underset{x\to 3}{lim}\frac{\frac{1}{x^2+2x}-\frac{1}{15}}{x-3}}$
Computing the limit by simplifying the function as follows, ${\displaystyle \underset{x\to a}{lim}\frac{x^2-a^2}{\sqrt{x}-\sqrt{a}}=\underset{x\to a}{lim}\left((\sqrt{x}+\sqrt{a})(x+a)\right)}$
${\displaystyle \underset{x\to a}{lim}\frac{x^2-a^2}{\sqrt{x}-\sqrt{a}}=\underset{x\to a}{lim}\frac{(x-a)(x+a)}{\sqrt{x}-\sqrt{a}}}$
${\displaystyle =\underset{x\to a}{lim}\frac{({\left(\sqrt{x}\right)}^2-{\left(\sqrt{a}\right)}^2)(x+a)}{\sqrt{x}-\sqrt{a}}}$
${\displaystyle \underset{x\to a}{lim}\left((\sqrt{x}+\sqrt{x})(x+a)\right)}$
${\displaystyle \underset{x\to a}{lim}\frac{x^2-a^2}{\sqrt{x}-\sqrt{a}}=\underset{x\to a}{lim}\left((\sqrt{x}+\sqrt{a})(x+a)\right)}$
${\displaystyle =(\sqrt{a}+\sqrt{a})(a+a)=2\sqrt{a}\left(2a\right)}$
${\displaystyle =2a\sqrt{a}}$