Limits and Continuity

From an interval to an instant

Average speed uses an interval: change in position divided by elapsed time. At one instant, that quotient would be $0/0$. Calculus keeps the interval nonzero and asks what the average rates approach as the interval shrinks.

From $t=a$ to $t=a+h$, the average rate is $[s(a+h)-s(a)]/h$. If the secant slopes approach one finite value as $h\to0$, that value is $s'(a)$, the instantaneous rate and tangent slope. Its units are still output-units per input-unit.

A secant line becoming a tangent line

For $s(x)=x^2$, the secant from $x=1$ to $x=1+h$ has run $h$, rise $2h+h^2$, and slope $2+h$. As $h$ approaches zero, its slope approaches 2.

The interactive version of this figure is part of the full study guide.

Read the approach in the correct order

In $\lim_{x\to a}f(x)=L$, the input $x$ moves toward $a$ and the output $f(x)$ moves toward $L$. The notation describes nearby inputs, so it does not tell you what happens exactly at $x=a$.

A two-sided limit exists only when both one-sided limits equal the same number. One isolated function value cannot change that nearby trend. Here the quotient equals $x+1$ whenever $x\ne1$, so the limit is 2 even if $f(1)$ is missing or assigned somewhere else.

The limit and the assigned value are separate

The open point at $(1,2)$ records the nearby limit. A separate filled point represents the chosen value $f(1)$. Moving that filled point does not change the limit.

The interactive version of this figure is part of the full study guide.

Approach the target along the curve

Trace the graph toward the target from the left and record the height it approaches. Then repeat from the right. Ignore any filled point at the target until both approach values are known.

Matching sides give the two-sided limit. Different finite heights create a jump, so the two-sided limit does not exist. A graph gives an estimate unless exact coordinates or an algebraic rule confirm the value.

Approach the same input from both sides

As the distance from zero shrinks, the left branch approaches output 1 and the right branch approaches output 3. The unequal one-sided limits mean the two-sided limit does not exist.

The interactive version of this figure is part of the full study guide.

Build a table around the target

Choose paired inputs below and above the target, then move each pair closer. For $a=2$, the pairs $1.9,2.1$ and $1.99,2.01$ show both directions. A one-sided table cannot establish a two-sided limit.

Look for outputs stabilizing near one number, not becoming exactly equal to it. Here the outputs approach 4 while the target value is undefined. The table estimates the trend; simplifying the expression to $x+2$ for $x\ne2$ explains it.

Build a two-sided limit table

At one, two, and three decimal places, paired inputs approach 2 from below and above. Their outputs under the nearby rule $x+2$ approach 4 at the same rate.

The interactive version of this figure is part of the full study guide.

Break a complicated expression into known limits

Limit laws formalize a stable idea: addition and multiplication preserve nearby behavior. If f(x) approaches L and g(x) approaches M, then f(x)+g(x) approaches L+M, and f(x)g(x) approaches LM. Constant multiples, integer powers, and suitable roots work the same way.

The quotient law has an essential restriction: M cannot be zero. A denominator approaching zero can produce an infinite limit, a removable form, or no limit at all. The laws therefore tell you what substitution is allowed to conclude, not merely what arithmetic to perform.

Polynomials are continuous everywhere, so their limits come from direct substitution. Rational functions behave the same wherever the denominator is nonzero. These facts turn many multi-step-looking limits into a quick evaluation.

Limit laws preserve arithmetic

As x approaches 1, the component functions approach 2 and 3. Their sum therefore approaches 5; the graph makes the component limits and combined limit visible.

The interactive version of this figure is part of the full study guide.

Change the form, not the nearby function

Direct substitution is a diagnostic first step. A finite value finishes the problem for a continuous expression; 0/0 does not. It means numerator and denominator vanish together, so their common structure may be removable.

Factoring can expose a common factor. Canceling that factor is valid only where it is nonzero, which is enough for a limit because x approaches the target without equaling it. For radical expressions, multiplying by a conjugate plays the same role by creating a factor that cancels.

Keep the original exclusion visible. The simplified rule explains the nearby outputs and supplies the limit, but it does not retroactively define the original function at the hole.

Let the first result choose the next step

Direct substitution should usually come first because it reveals both easy limits and the reason a harder one is blocked. A finite result ends the calculation. A nonzero number divided by zero calls for one-sided sign analysis. The form 0/0 invites algebraic simplification rather than an answer.

Match the expression to its structure. Factor polynomials, rationalize radicals, combine complex fractions, use known trigonometric limits, or compare bounding functions. When the input approaches infinity, divide by the dominant power or reason from relative growth.

Graphs and tables are appropriate when the problem provides them, but they support estimates rather than hidden exact arithmetic. The best method is the one whose conditions you can state and whose conclusion addresses the requested side or end behavior.

The indeterminate form tells you what to try next

Direct substitution at x=2 gives 0/0. The hole and the visible line indicate that factoring—not declaring the limit undefined—is the useful next move.

The interactive version of this figure is part of the full study guide.

Control magnitude instead of tracking every oscillation

Some functions are too erratic for direct substitution or ordinary algebra. The Squeeze Theorem replaces the exact output with bounds. If $g(x)≤f(x)≤h(x)$ near a, and both outside functions approach L, there is no vertical room for f(x) to approach anything else.

A useful bound often comes from absolute value. Since $|\sin(1/x)|≤1$, multiplying by $x^2$ gives $-x^2≤x^2\sin(1/x)≤x^2$. The sine term oscillates infinitely often near zero, but its amplitude collapses inside the two parabolas.

The inequality only needs to hold near the target, and the outside limits must be equal. Bounds approaching different numbers merely confine possible outputs; they do not determine one limit.

A trapped function shares the outside limit

The oscillating curve x²sin(1/x) remains between −x² and x². Both bounds approach zero, forcing the middle function to approach zero too.

The interactive version of this figure is part of the full study guide.

Preserve the same mathematical claim

A limit statement has two coordinated movements: x approaches a, and f(x) approaches L. A graph shows that movement spatially, a table samples it numerically, and an equation explains it symbolically. None of these changes what “approach” means.

For $(x^2-1)/(x+1)$ near x=−1, the graph approaches the hole at y=−2. Table values on either side close in on −2. Factoring produces x−1 for all nearby allowed inputs, which also approaches −2. The three views agree while the original function remains undefined at the target.

When representations conflict, check scale, rounding, copied values, and domain first. A graph with a coarse window and a table with distant inputs may provide weaker evidence than exact algebra.

Graph, table, and expression must describe the same approach

The graph shows a hole at x=−1, the table closes in on −2, and factoring gives the nearby rule x−1. Agreement across all three representations supports the same limit.

The interactive version of this figure is part of the full study guide.

Diagnose the failure before trying to repair it

A discontinuity is not simply a break in a drawing. Its type comes from limits. At a removable discontinuity, both sides approach the same finite value, but the function is missing there or assigns a different value. One point can repair it.

At a jump, the left and right limits are finite but unequal. No single definition at the target can make both sides agree. At an infinite discontinuity, at least one side grows without bound, often along a vertical asymptote. The signs may differ across the asymptote.

Oscillatory behavior is another way a limit can fail: outputs keep moving through a range without approaching one value. Naming the failure guides later work with piecewise functions, integrals, and theorem hypotheses.

Holes, jumps, and vertical asymptotes fail continuity differently

The blue curve has a removable hole at x=−2. The red reciprocal branch becomes unbounded near x=1. These failures require different limit statements.

The interactive version of this figure is part of the full study guide.

Continuity joins value and approach

A function is continuous at x=a when its local trend and assigned value agree. First, f(a) must be defined. Second, the left and right limits must agree so that $\lim_{x\to a}f(x)$ exists. Third, that common limit must equal f(a).

The order matters. If the one-sided limits disagree, changing f(a) cannot fix the jump. If the limit exists but f(a) is missing or different, the discontinuity is removable because one assignment can satisfy the final equality.

Continuous functions allow direct substitution for limits and support major theorems later in calculus. Always state the interval or point under discussion; a function can be continuous almost everywhere while failing at a few inputs.

Use only directions contained in the domain

To be continuous on an open interval, a function must be continuous at every point inside it. On a closed interval [a,b], the interior condition remains the same, but each endpoint has domain on only one side.

At the left endpoint, require $\lim_{x\to a^+}f(x)=f(a)$. At the right endpoint, require $\lim_{x\to b^-}f(x)=f(b)$. Asking for the missing outside side would test behavior where the restricted function is not defined.

Polynomials are continuous on every real interval. Rational functions are continuous wherever their denominator is nonzero, and compositions inherit continuity when each stage is defined. For piecewise functions, inspect every joining input separately.

Fill a hole with the value the curve already approaches

A removable discontinuity contains enough nearby information to determine its repair. Compute $L=\lim_{x\to a}f(x)$. If L is finite, define or redefine f(a)=L. The new function then has a value, a limit, and equality between them.

For $(x^2+x-2)/(x-1)$, factoring gives $(x+2)(x-1)/(x-1)=x+2$ for x≠1. Nearby outputs approach 3, so f(1)=3 is the unique assignment that restores continuity.

This method cannot repair a jump, vertical asymptote, or oscillation because those behaviors lack one finite two-sided limit. Altering a single point never changes either one-sided trend.

Choose the missing value from the limit

The nearby rule approaches 3 at x=1. Defining f(1)=3 fills the hole; any other assigned value leaves a discontinuity.

The interactive version of this figure is part of the full study guide.

Unbounded behavior still has direction

The notation $f(x)\to+\infty$ means outputs exceed every positive bound as x approaches the target; it is not a finite limit equal to a number called infinity. Likewise, $-\infty$ records unbounded negative behavior.

For $1/(x-1)$, a small negative denominator on the left produces a large negative quotient, while a small positive denominator on the right produces a large positive quotient. Thus the two one-sided infinite limits have opposite signs.

A denominator zero must remain after simplification to support a vertical asymptote. If its factor cancels, the behavior is usually a hole. For higher powers, parity helps: an even-power denominator often gives the same sign on both sides; an odd power changes sign.

At large magnitude, dominant terms decide the ratio

A limit at infinity asks what happens as x moves indefinitely far left or right. It does not describe a reachable endpoint. For rational functions, divide numerator and denominator by the largest relevant power of x so lower-degree terms visibly shrink toward zero.

If the numerator degree is smaller, the ratio approaches zero. Equal degrees produce the ratio of leading coefficients. If the numerator degree is larger, the function does not have a finite horizontal asymptote, though polynomial division may reveal slant or polynomial end behavior.

Horizontal asymptotes describe end behavior, not barriers. A function may cross y=L many times and still approach L as x tends to infinity.

Leading terms control rational end behavior

For (2x²−1)/(x²+3), lower-degree terms become negligible as |x| grows. The ratio approaches 2 in both directions, producing the horizontal asymptote y=2.

The interactive version of this figure is part of the full study guide.

Continuity prevents skipped output levels

Suppose f is continuous on [a,b] and N lies between f(a) and f(b). The Intermediate Value Theorem guarantees at least one c in [a,b] with f(c)=N. If N differs from both endpoint values, then c lies in the open interval.

The theorem proves existence, not location, uniqueness, or an exact formula. A wavy continuous function may cross the target level several times. Numerical methods can then approximate a crossing, but continuity is what guarantees one is available to find.

Every hypothesis matters. A jump can move from below N to above N without ever equaling N. State continuity on the full closed interval, compute the endpoint values, and show that the target is between them.

Continuity guarantees a crossing, not its location

The continuous cubic is below 2 at x=0 and above 2 at x=2. The Intermediate Value Theorem guarantees at least one c in (0,2) with f(c)=2.

The interactive version of this figure is part of the full study guide.