How do you find the limit of a function?

To find the limit of a function as ( x ) approaches a value ( a ), we ask: "What value does ( f(x) ) get close to as ( x ) gets close to ( a )?"

Steps:

1. Plug in ( a ): Try substituting ( x = a ) into ( f(x) ).
2. Simplify, if needed: If direct substitution gives an undefined form (like 0/0), simplify the function (e.g., factor or rationalize) and try again.
3. Use limit laws: For tricky cases, apply limit properties or special techniques like L'Hôpital's Rule if you're studying calculus.

Example:
Find ( lim_{x to 2} frac{x^2 - 4}{x - 2} ).

4. Plug in ( x = 2 ): ( frac{2^2 - 4}{2 - 2} = frac{0}{0} ) (undefined).
5. Factor numerator: ( x^2 - 4 = (x-2)(x+2) ).
6. Simplify: ( frac{(x-2)(x+2)}{x-2} = x+2 ) (for ( x ne 2 )).
7. Plug in ( x = 2 ): ( 2 + 2 = 4 ).

So, the limit is 4.

Takeaways:

• Substitute first, then simplify if you get 0/0 or undefined.
• Limits reveal a function's behavior as ( x ) nears a value, not necessarily at that value.