Limit Laws in Action

Calculus 1 · Axiom Academy

Break a complicated limit into pieces you can already evaluate — then recombine. Watch each law do the splitting. Suppose we know and . The limit laws say a limit "respects" algebra: wherever the expression has a seam — a plus, a times, a divide — you may cut there, take the limit of each side, and recombine. Cut the complex limit at its + seam Evaluate each piece, then recombine Here both pieces approach finite values as , so their heights simply stack. Watch the two sub-limits grow to 5 and 4 , then add into a single bar of height 9 . A product reads as area. Let one factor be the width and the other the height of a rectangle. As the width settles at 4 and the height at 3 , so the rectangle fills to an area of . Split numerator from denominator and limit each. The numerator heads to 10 , the denominator to 2 . The one condition you must check first: the denominator's limit is , so dividing is legal and the answer is 10/2 = 5 . A power is just repeated multiplication, so the product law applies as many times as the exponent. The base (x+1) approaches 3 as ; cubing copies that limit three times: . The power shows up when you chain the laws. Take one genuinely complex limit and peel it from the outside in — quotient first, then sum, then power and constant-multiple on each term — until every leaf is something you can read off by substitution. Quotient law splits it: (denominator ). Power + constant-multiple: 3(2)^2 + 4(2) = 12 + 8 = 20 ; bottom = 2+1 = 3 .

This is the written version of the interactive lesson above. See the full Calculus 1 course.