CLEP Calculus offers students the opportunity to demonstrate their understanding of calculus concepts and applications, enabling them to earn college credit.
Limits form the foundation of calculus, allowing us to analyze what happens to a function as the input approaches a particular value. Think of a limit as asking, “What value does \( f(x) \) get close to as \( x \) approaches some number?”
If \( \lim_{x \to a} f(x) = L \), it means as \( x \) gets closer to \( a \), \( f(x) \) gets closer to \( L \). This is crucial for defining derivatives and integrals.
A function is continuous at a point if its limit at that point exists and equals the function’s value there. In simple terms, a continuous function can be drawn without lifting your pencil!
Limits help us understand behaviors like holes, jumps, and asymptotes in graphs, and allow us to define instantaneous rates of change.
Finding \( \lim_{x \to 2} (x^2 - 3) \) by plugging in \( x = 2 \) gives \( 4 - 3 = 1 \).
A function with a hole at \( x = 1 \) might still have \( \lim_{x \to 1} f(x) \), even if \( f(1) \) is undefined.
Limits are about what a function approaches, not necessarily what it equals at a point.