5.3 Limits, Continuity, and Rates of Change

Why This Topic Matters

The AEPA differential-calculus competency includes evaluating limits, demonstrating continuity, and analyzing derivatives through tangent slopes and difference quotients. Those ideas often appear before a formal derivative rule is needed. You may see a graph with a hole, a table approaching a value, a piecewise function, or a rate context asking whether a calculation is average or instantaneous.

A limit asks what value f(x) approaches as x approaches a number. It does not first ask whether f(a) is defined. For example, if f(x) = (x^2 - 9)/(x - 3), direct substitution at x = 3 is undefined, but factoring gives (x - 3)(x + 3)/(x - 3). For x not equal to 3, the expression behaves like x + 3, so the limit as x approaches 3 is 6. The graph has a hole at x = 3 unless the function is separately redefined there.

Limit Types

One-sided limits are essential. The two-sided limit exists only when the left-hand and right-hand limits agree. A graph can have a filled dot at one y-value and an open circle at another. The filled dot gives f(a); the open-circle approach may give the limit. A common student misconception is to read only the filled dot and ignore the graph approaching from both sides.

Continuity

A function is continuous at x = a when three conditions all hold: f(a) is defined, the limit as x approaches a exists, and the limit equals f(a). For a piecewise function, start by finding the boundary value from each side. Suppose f(x) = x + 2 for x < 1 and f(x) = kx^2 for x >= 1. The left-hand limit at 1 is 3. The right-hand limit and function value are k. Continuity requires k = 3.

This is also a teaching issue. A student may think "both pieces are continuous, so the piecewise function is continuous." Each formula may be continuous on its own interval, but the boundary still has to connect. Ask what the graph does as x approaches the boundary from the left and from the right.

Rates Of Change

The average rate of change of f from x = a to x = b is (f(b) - f(a))/(b - a). It is the slope of the secant line and carries units: output units per input unit. If h(t) is height in feet and t is time in seconds, average rate is feet per second.

The instantaneous rate of change is the limiting slope at a point. It is defined by the difference quotient limit: lim as h approaches 0 of [f(a + h) - f(a)]/h. AEPA may give this expression and expect you to recognize it as f'(a), which is faster than expanding everything. For f(x) = x^2 + 3x, the expression [f(2 + h) - f(2)]/h approaches 7, matching f'(2) = 2(2) + 3.

Worked Reading Example

A table shows f(1.9) = 4.81, f(1.99) = 4.9801, f(2.01) = 5.0201, and f(2.1) = 5.21. These values suggest the limit near x = 2 is 5, even if the table does not state f(2). A responsible conclusion is "the evidence supports a limit of 5," not "the function value is definitely 5." If the problem gives f(2) = 8, then the limit can still be 5 and the function is not continuous at 2.

Common Traps

• Substitution trap: getting 0/0 means simplify or use another method, not that the limit is 0.
• Function-value trap: f(a) can differ from the limit.
• One-sided trap: a two-sided limit fails if the two sides approach different values.
• Unit trap: rates need units, and changing input units changes the rate.
• Secant/tangent trap: average rate over an interval is not automatically the instantaneous rate at an endpoint.

For the exam, annotate what is being asked: value, limit, continuity, average rate, or instantaneous rate. That single label often determines the method before any algebra begins.

Algebraic Limit Moves

Common algebraic tools include factoring, rationalizing, combining fractions, and using known standard limits when appropriate. For a radical limit such as lim as x approaches 0 of (sqrt(x + 9) - 3)/x, multiply by the conjugate. The numerator becomes x, and the expression simplifies to 1/(sqrt(x + 9) + 3), so the limit is 1/6. The key is that x is canceled only after the expression is rewritten for nearby x-values, not by pretending x equals zero.

For infinite behavior, describe the trend rather than forcing a finite number. If f(x) = 1/(x - 2)^2, both sides grow positive without bound as x approaches 2, so the graph has a vertical asymptote. If f(x) = 1/(x - 2), the left side goes negative without bound and the right side positive without bound; the two-sided finite limit does not exist. A teacher-certification explanation should name the one-sided behaviors because students often summarize both cases as simply "undefined."