Computing Limits

Pre-Calculus · Axiom Academy

Three worked limits, each cracked open with a different algebraic technique Direct substitution gives — an indeterminate form. The numerator is a difference of squares, so we factor and cancel. The graph of is the line y=x+3 with a hole at x=3 . The curve heads straight for height 6 . Substituting x=4 again gives . A square root in the numerator is the cue to multiply by the conjugate. Away from x=4 the function simplifies to . As the height closes in on . Problem 3 — Limits at Infinity There is nothing to cancel here. For a limit as , divide every term by the highest power of x and see what survives. As x grows, flattens against the horizontal asymptote y=3 . Nice work. A limit that gives isn't undefined — it's an invitation to rewrite the expression. Match the algebra to the obstacle: Factoring: when a polynomial shares a factor top and bottom, factor and cancel it, then substitute. Here . Rationalizing: when a square root causes the , multiply by the conjugate to clear it, then cancel. Here the limit is . Divide by the highest power: for a limit as , divide every term by the largest power of x ; the leftover constant ratio is the answer. Here it is . Always try direct substitution first — only reach for these techniques when it produces an indeterminate form. The graph is your sanity check: a hole sits exactly at the limit value, and a horizontal asymptote is the limit at infinity.

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