To find the limit of a function ( f(x) ) as ( x ) gets close to a certain value ( a ), we can follow some simple steps. This is shown as ( \lim_{x \to a} f(x) ).

1. Direct Substitution:

   • First, try plugging in ( a ) into the function. Find ( f(a) ).
   • If ( f(a) ) gives us a number (and it's not too big or small), then we can say ( \lim_{x \to a} f(x) = f(a) ).

2. Indeterminate Forms:

   • If putting in ( a ) gives an unclear result, like ( \frac{0}{0} ) or ( \frac{\infty}{\infty} ), we need to try something else.

3. Factoring:

   • Try to break down the expression to make it easier. For example, if we have ( f(x) = \frac{x^2 - a^2}{x - a} ), we can factor it to ( \frac{(x - a)(x + a)}{x - a} ). This can be simplified to ( x + a ) when ( x ) is not equal to ( a ).

4. Rationalization:

   • If the function has a square root, multiply by a special form called the conjugate to make it simpler.

5. Numerical (Table of Values):

   • Sometimes, it helps to make a table with values that get closer to ( a ) from both sides. This can show us how the limit behaves.

6. Graphical Interpretation:

   • Drawing a graph of the function can also help us see what happens to the function as it gets near ( a ).

Conclusion: If none of these steps help us find the limit, we might need to consider special limits (like when ( x ) goes to infinity) or use L'Hôpital's Rule.