Module 06: Limits: nearby behavior, not point substitution

A. Prerequisites and observable goals#

B. The central idea: separate “near” from “at”#

The notation $\lim_{x\to a}f(x)=L$ describes the output values for inputs arbitrarily close to $a$, excluding $a$ itself. It does not assert $f(a)=L$, nor even that $f(a)$ exists. A function can have a hole, a reassigned value, or no definition at the approached point while nearby values approach a well-defined number.

A left-hand limit uses inputs $x<a$; a right-hand limit uses inputs $x>a$. A two-sided finite limit exists exactly when the two one-sided limits exist and equal the same finite number. A table can suggest an approach, and a graph can communicate the pattern, but neither a finite table nor a coarse screen proves the behavior for all sufficiently close inputs. Precise definitions and limit laws: [O1L].

C. Limit laws and forms that require work#

If $f(x)\to A$ and $g(x)\to B$ as $x\to a$, then sums, products, and constant multiples have the expected limits. The quotient limit is $A/B$ only when $B\ne0$. Polynomial limits may be found by substitution, and rational functions may be evaluated by substitution wherever their denominator is nonzero.

The radian limits $\lim_{u\to0}\sin u/u=1$ and $\lim_{u\to0}(1-\cos u)/u=0$ are basic trigonometric limits. The first can be established by comparing sector and triangle areas, giving $\cos u\le\sin u/u\le1$ for small positive $u$, then using symmetry for negative $u$. The squeeze theorem says that if $g\le f\le h$ near $a$ and both outside functions tend to $L$, then $f$ also tends to $L$.

The statement $f(x)\to+\infty$ describes unbounded growth, not convergence to a real number. A vertical asymptote $x=a$ occurs when at least one one-sided function value tends to $+\infty$ or $-\infty$. A horizontal asymptote $y=L$ describes a finite limit as $x\to+\infty$ or $x\to-\infty$. A graph may cross a horizontal asymptote.

D. Why canceling near a hole is legitimate#

For $x\ne3$, $(x^2-9)/(x-3)=x+3$. Limits ignore the value at the approached input, so both expressions have the same limit as $x\to3$, namely $6$. This is not cancellation at zero: every nearby input considered in the quotient has $x-3\ne0$. The limit is then taken for the equivalent nearby expression.

At infinity, divide a rational function’s numerator and denominator by the highest relevant power of $x$. For example,

For radicals, $\sqrt{x^2}=|x|$ matters: $\sqrt{x^2+1}/x$ tends to $1$ at $+\infty$ but $-1$ at $-\infty$.

E. Choose the method before calculating#

Try substitution where the usual continuity rules apply. If it produces $0/0$, inspect for factors, a conjugate, or a standard trigonometric form. If the denominator approaches zero but the numerator does not, examine signs on each side. For absolute values or piecewise formulas, split into sides. At infinity, compare growth or divide by powers. If an oscillatory factor is bounded and multiplied by something approaching zero, try squeezing. Do not use L’Hôpital’s rule before its hypotheses and derivative machinery have been taught in Module 15.

F. Worked examples#

A radical. For $x\ne4$,

The conjugate converts the numerator to $x-4$, which is a cancellable nonzero factor near $4$.

A trigonometric rescaling. Write

As $x\to0$, the second factor tends to $1$, so the limit is $5/2$. The variable inside sine must match the denominator of the standard limit.

One-sided sign. As $x\to2^+$, $1/(x-2)$ tends to $+\infty$; from the left it tends to $-\infty$. There is no common two-sided finite limit. By contrast $1/(x-2)^2$ tends to $+\infty$ from both sides.

G. Independent practice#

F foundation C standard course work T transfer S optional stretch L runnable lab or project. Attempt the complete set before opening its solutions.

H. Complete solutions: Module 06#

Compare the first step where your work differs. A corrected answer is not yet an independent success.

Open every solution

I. Retention and readiness#

Revisit M01.03, M04.03, and M05.03. Explain what $0/0$ tells you: the current form is inconclusive, not that the answer is zero. Advance when you can choose among factoring, conjugates, one-sided cases, and squeezing in a mixed set. Review R1 follows Module 08.