Limits Explained Visually | Why Calculus Needs Limits - chapter 08

Why do we need limits if we can sometimes just substitute the value? In this chapter of The Maths Academy Calculus Complete Master Course, we build the real intuition behind limits. We begin with a simple function, f(x) = 2x + 1, where direct substitution works. Then we see why limits become necessary when direct substitution fails: missing points, 0/0 forms, jump discontinuities, oscillation, unbounded behavior, and the foundation of derivatives. This chapter is designed for beginners who want to understand calculus visually, slowly, and deeply. You will learn: • What a limit really means • Why “approaching” matters in calculus • Left-hand and right-hand limits • Input and output behavior • Function value vs limiting value • Why holes can still have limits • Why 0/0 does not always mean “no answer” • Why jump discontinuities may destroy a limit • Why oscillation and unbounded behavior are different problems • Why derivatives need limits This is Chapter 7 of The Maths Academy Calculus Complete Master Course. The Maths Academy — cinematic mathematics, explained from the ground up. #Calculus #Limits #Mathematics #TheMathsAcademy #LearnCalculus