Determining Limits Using Algebraic Manipulation

Plugging in a number and getting 0/0 doesn't mean the limit fails to exist. It means the expression is hiding a common factor that vanishes at that exact x-value, creating a hole in the graph rather than a jump or asymptote. Your job is to rewrite the expression so that factor is gone, then substitute again.

For rational expressions, factor the numerator and denominator and cancel whatever matches. For expressions with square roots, multiply top and bottom by the conjugate of the radical part, which uses the difference of squares to turn a square root into a plain variable that can cancel with the denominator. Either way, the algebra doesn't change what the function approaches, it just removes the one point where substitution was blocked.

This is the main algebraic toolkit before limits get harder: it sets up the idea that a limit only cares about behavior near a point, not at it, which is exactly what makes continuity meaningful later. When factoring and conjugates don't resolve the indeterminate form, later topics bring in more powerful tools like L'Hopital's rule, but for now, these two moves handle the overwhelming majority of 0/0 limits you'll see.