Evaluate limit of a functon at a point Suppose we want to evaluate the limit of  a function $f(x) = 2x + 3$ at $2$. One way is to to find $f(2)$ and declare it to be the result. Is it a correct step to do since from the definition of limit we know that we are interested in the value when $x$ approaches close to 2 but not exactly at 2. In this sense, replacing $x$ by 2 might not be a correct step.
 A: Let's consider a slightly more interesting (but related) example. Let $$g(x)=\frac{2x^2-x-6}{x-2},$$ defined for all real $x\ne 2.$ It can be shown that for all $x\ne 2,$ we have $g(x)=2x-3.$ Of course, it doesn't make sense to talk about $g(2),$ since it is undefined, there. However, we can still talk about the limit as $x\to 2$. Noting that for all $x\ne 2$ we have $$|g(x)-1|=|(2x-3)-1|=|2x-4|=2|x-2|,$$ then for any $\epsilon>0$ and any $x\in\Bbb R$ such that $0<|x-2|<\frac\epsilon2,$ we have $|g(x)-1|<\epsilon,$ so $\lim_{x\to 2}g(x)=1.$
A: I think it is better to stick to the rules of limits.
$\displaystyle \begin{aligned}\lim_{x \to 2}f(x) &= \lim_{x \to 2}(2x + 3)\\
&= \lim_{x \to 2}2x + \lim_{x \to 2}3\\
&= 2\lim_{x \to 2}x + 3\\
&= 2\cdot 2 + 3 = 7\end{aligned}$
We have used the fact that $\lim_{x \to a}x = a$ which can be verified almost trivially using definition of a limit.
I find it totally unnecessary to invoke any higher concepts (other than definition of limits and rules of limits) to evaluate limits. Only when it is not possible to solve a problem using basic definitions and rules I go for the high level techniques like continuity, L'Hospital, series expansion, Stirling formula and what not.
We should try to stick to simple ideas to avoid confusion and learn to understand the powers of these simple rules of limits. Using these rules of limit we can show that limit of any expression consisting of algebraic, trigonometric, logarithmic, exponential functions when $x \to a$ can be evaluated by a direct substitution of $x = a$ in the expression provided that such a substitution does not lead to any undefined expression (like zero in denominator, log of zero etc.)
