Continuity and Limits Review

When we say a function is continuous at a point, we mean the function's graph doesn't have any breaks, jumps, or holes at that point. Think of it as drawing without lifting your pencil!

For a function to be continuous at a point x = c, three key conditions must be met:

1. The function value f(c) must exist (the function is defined at that point)
2. The limit of f(x) as x approaches c must exist
3. The limit must equal the function value: lim f(x) = f(c)

💡 Quick Tip: When checking continuity, think "Does, Exists, Equals" - Does the function exist at the point? Does the limit exist? Do they equal each other?

Horizontal asymptotes are connected to limits - they occur when a function approaches a specific y-value as x gets very large (positive or negative). This happens when lim f(x) = L as x approaches infinity.

Remember that trigonometric functions like sin(x) and cos(x) are continuous everywhere, while rational functions might have points of discontinuity where denominators equal zero.