Let us take an example.

Example: Lim_{x-> 5} (x² - 25)/(x² + x - 30)

Now, if we directly put x = 5, then we get

(5² - 25)/(5² + 5 - 30) = (25 - 25)/(25 + 5 - 30) = 0/0, which is not defined.

Now, Lim_{x-> 5} (x² - 25)/(x² + x - 30) = Lim_{x-> 5} {(x - 5)*(x + 5)}/{(x - 5)*(x + 6)}

                                                         = Lim_{x-> 5} (x + 5)/(x + 6)

                                                         = (5 + 5)/(5 + 6)

                                                         = 10/11

In this way, we put the limit.

There are many other methods like L. Hospital Rule, etc. which solves the limit problems.