Grades Pre-Calculus & Calculus Limits & ContinuityThe Concept of a Limit

The Concept of a Limit

🎯 Limits & Continuity

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The Concept of a Limit

Understand what it means for a function to approach a value as x approaches a point.

The limit of f(x) as x approaches a is L — written lim_{x→a} f(x) = L — if f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but not equal to a). The function need not be defined at x = a for the limit to exist.

\lim_{x \to a} f(x) = L \iff \text{for every } \varepsilon > 0,\ \exists\,\delta > 0 \text{ s.t. } 0 < |x-a| < \delta \implies |f(x)-L| < \varepsilon

In practice, we evaluate limits by direct substitution, factoring and canceling, rationalizing, or applying limit laws. Many limits that appear indeterminate (0/0) can be resolved by algebraic manipulation.

\lim_{x\to 2}\frac{x^2-4}{x-2} = \lim_{x\to 2}\frac{(x-2)(x+2)}{x-2} = \lim_{x\to 2}(x+2) = 4

Limit Laws

Sum: lim[f + g] = lim f + lim gProduct: lim[f · g] = lim f · lim gQuotient: lim[f/g] = lim f / lim g (lim g ≠ 0)Power: lim[f(x)^n] = (lim f(x))^nConstant: lim c = c

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Remember This!

If direct substitution gives 0/0, factor the numerator and denominator and cancel the common factor that is causing the issue. The resulting expression can be evaluated by substitution.

✏️ Try It!

What is lim_{x→3} (x² − 9)/(x − 3)?

Practice with CalcVerse

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