Limits (intro to calculus)
The idea
A limit describes where a function's outputs are heading as the input approaches a target — whether or not the function ever arrives there. Writing that the limit of f(x) as x approaches 4 equals 1/6 claims f(x) gets arbitrarily close to 1/6 as x closes in on 4 from both sides. This is the gateway idea of calculus: the slope of a curve at a point and the exact area under a curve are both defined as limits. Crucially, a limit cares only about the approach, never about the value at the target itself, which may be different or missing entirely.
Many limits yield to direct substitution, but the interesting ones land on 0/0 — an indeterminate form, meaning the form alone decides nothing and the expression must be rewritten. Factoring and canceling, or multiplying by a conjugate when square roots appear, removes the shared factor that causes the 0/0 and exposes the true limiting value. The misconception to abandon: 0/0 is not 0, not 1, and not automatically undefined — different functions wearing the 0/0 form approach genuinely different limits, which is exactly why the algebra has to dig deeper.
Worked example
Evaluate the limit of (√(x + 5) − 3)/(x − 4) as x approaches 4, and support the answer numerically.
- Try substitution first: at x = 4 the numerator is √9 − 3 = 0 and the denominator is 0 — the indeterminate form 0/0, a signal to rewrite rather than quit.
- Multiply the numerator and denominator by the conjugate √(x + 5) + 3. The numerator becomes (x + 5) − 9 = x − 4, by the difference-of-squares pattern (p − q)(p + q) = p² − q².
- The expression is now (x − 4)/((x − 4)(√(x + 5) + 3)). For x ≠ 4 the common factor cancels, leaving 1/(√(x + 5) + 3) — legal because the limit ignores the point x = 4 itself.
- Substitute into the simplified form: 1/(√9 + 3) = 1/6 ≈ 0.1667.
- Numerical support from both sides: at x = 4.01 the original expression evaluates to about 0.16662, and at x = 3.99 to about 0.16672 — both closing in on 1/6.
Answer. The limit is 1/6, approximately 0.1667.
Check your understanding
- Why is 0/0 called indeterminate rather than undefined, and what does that distinction tell you to do next?
- How can a function have a limit at a point where it is not even defined?
- Why is canceling the factor x − 4 legitimate inside a limit when the cancellation changes the function at that one point?
- How do one-sided limits sharpen the picture when a function behaves differently on the two sides of a point?
Build the foundations first
Limits (intro to calculus) builds on these concepts. If any feel shaky, start there.