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MathidiaMathematics Academy
Calculus foundations
Lesson 6 of 75
Introduction

Limits and asymptotic behaviour

Objective. Evaluate and interpret finite and infinite limits.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The limit of a function f(x) as x approaches a describes the value f(x) approaches, written lim(x→a) f(x)=L, formalised by the ε–δ condition that |f(x)−L| can be made arbitrarily small by keeping x sufficiently close to a. Asymptotic behaviour studies limits as x→±∞ or where f blows up, giving horizontal and vertical asymptotes. For f(x)=(2x²+1)/(x²+3), as x→∞ divide numerator and denominator by x² to get lim=2, so y=2 is a horizontal asymptote. Limits that produce 0/0 or ∞/∞ require algebraic manipulation such as factoring or dividing by the highest power before substituting. Recognising which indeterminate form arises is the first step in any limit calculation.

Worked example

Evaluate the limit lim(x→2) (x²−4)/(x−2).

2 … 4
Illustrated card showing 2 … 4
  1. 1Direct substitution gives 0/0, an indeterminate form, so factor instead.
  2. 2Factor the numerator: x²−4=(x−2)(x+2).
  3. 3Cancel the common factor (x−2), leaving x+2 for x≠2.
  4. 4Substitute x=2 into x+2 to get the limit value 4.

Answer: 4

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Common mistake. Students substitute x=a directly even when it produces 0/0 and wrongly conclude the limit is undefined; instead they must factor, cancel common terms, or rationalise before substituting again.

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