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Unit 6 · Bridge to calculus

Limits and Continuity

The bridge into calculus. Limits are found three ways, continuity is defined through them, and an average rate of change becomes an instantaneous one.

What a strong answer looks like

Unit 6 answers should separate the value a function approaches from the value it takes. Nearly every question in this unit turns on that distinction.

Practice this unitCourse overview

Topics in this unit

1

What a Limit Actually Describes

Know

A limit describes where the outputs are heading as the inputs approach a value, deliberately ignoring what happens at the value itself. The function need not be defined there, and if it is, the value can differ from the limit.

Apply

Read a limit from a graph by tracing toward the point from each side, or from a table by watching the outputs stabilize. Both are legitimate methods, not fallbacks.

Watch out

Reporting the function value when the limit was asked for is the defining error of this unit, and it is exactly what a removable discontinuity is built to expose.

Study move

For a graph with a hole and a separately plotted point, state the limit and the function value separately and explain why they differ.

2

One-Sided Limits

Know

A two-sided limit exists only when both one-sided limits exist and agree. Piecewise functions and jumps are where they part company.

Apply

At a piecewise boundary, evaluate each branch as it approaches the boundary and compare. Equal values mean the limit exists; different values mean it does not.

Watch out

When the one-sided limits disagree, the answer is that the limit does not exist. It is not the average of the two, and it is not whichever branch is defined at the point.

Study move

Given a piecewise function, compute both one-sided limits at each boundary and decide continuity there before looking at the rest of the graph.

3

Algebraic Techniques

Know

When substitution produces 0/0, the expression is indeterminate rather than undefined. The form is an instruction to rewrite, not a verdict.

Apply

Factor and cancel when the expression is rational, and multiply by the conjugate when a radical creates the difficulty. Then substitute into the simplified form.

Watch out

Only 0/0 and similar indeterminate forms call for this work. A nonzero number over zero is not indeterminate; it signals unbounded behavior instead.

Study move

Sort a mixed set of limits by which technique each requires before evaluating any of them.

4

Infinite Limits and Limits at Infinity

Know

These are two different questions. An infinite limit asks what happens near a vertical asymptote; a limit at infinity asks about end behavior far from the origin.

Apply

Near an asymptote, determine the sign of the denominator from each side to decide the direction. At infinity, divide every term by the highest power in the denominator and let the small terms vanish.

Watch out

The two sides of a vertical asymptote often go in opposite directions, so a one-sided answer is required unless both sides agree.

Study move

For one rational function, answer all four questions: the two one-sided limits at its asymptote and the two limits at positive and negative infinity.

5

Continuity and Rates of Change

Know

A function is continuous at a point when the limit exists, the function is defined there, and the two agree. Failing that in different ways gives removable, jump, and infinite discontinuities.

Apply

To classify a discontinuity, ask whether the limit exists. A finite limit at an undefined point is removable, disagreeing one-sided limits give a jump, and unbounded behavior gives an infinite discontinuity.

Watch out

A removable discontinuity is still a discontinuity. The function is not continuous there until the value is actually redefined.

Study move

Take an average rate of change over an interval anchored at a point, simplify it, and let the interval shrink. Say in words what the resulting number measures.

Emphasized in this unit

Connections and techniques that receive extra attention in this honors sequence.

  • • Limits reached from a table as a first-class method, not a fallback
  • • Connecting a rational function asymptote to an infinite limit
  • • Average against instantaneous rate of change as the definition to carry forward

Varies by course

Related topics some schools attach to this unit and others leave out. Covered on request rather than assumed.

  • Derivative rules and introductory integration. Some accelerated courses continue into differentiation or integration. This course stops at limits, continuity, and instantaneous rates of change.

Mastery checklist

  • • Can state a limit from a graph or a table and distinguish it from the function value.
  • • Can compute one-sided limits at a piecewise boundary and decide whether the limit exists.
  • • Can recognize an indeterminate form and choose factoring or a conjugate accordingly.
  • • Can determine behavior near a vertical asymptote from each side.
  • • Can evaluate a limit at infinity by comparing degrees.
  • • Can classify a discontinuity and find a value that removes a removable one.
  • • Can explain how an average rate of change becomes an instantaneous rate of change.

Check yourself

  • • How can a limit exist at a point where the function is undefined?
  • • Why is 0/0 an instruction rather than an answer?
  • • What is the difference between a removable discontinuity and a jump discontinuity?

Modeling drill

An object has position s(t) = t² + 3t. Compute the average rate of change on the interval from 2 to 2 + h, simplify it, and describe what happens to that expression as h shrinks. Say what the resulting number measures physically.

limitone-sided limitindeterminate formconjugateinfinite limitlimit at infinitycontinuityremovable discontinuityjump discontinuityaverage rate of changeinstantaneous rate of change

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Unit 5: Conic Sections