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Unit 1
10–15% of examCalculus is the math of change, and limits are what make it work. In this unit you learn what it means for a function to approach a value, how to find limits from graphs, tables and algebra, and how limits decide whether a function is continuous. Derivatives and integrals are both built on these ideas.
Longer videos that cover the whole unit. Good for a first pass or a final review.
An average rate of change uses two points, but calculus asks how fast something is changing at one single instant. You get there by finding average rates over smaller and smaller intervals and seeing what value they approach.
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lim (x→c) f(x) = L means f(x) gets as close as you like to L when x is close enough to c, without x being equal to c. Because a limit is about values near c, it can exist even when f(c) is undefined or is a different number.
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To read a limit from a graph, follow the curve toward x = c from the left and from the right. The limit exists only if both sides head to the same value; a jump, a vertical asymptote or endless wiggling (oscillation) are common reasons it does not.
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A table of x-values creeping closer to c from both sides suggests what the limit is. A table is evidence, not proof, so check that the outputs from each side settle toward the same number.
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The limit of a sum, difference, product, constant multiple or quotient can be split into the limits of its parts (for a quotient, the bottom limit can't be 0). For polynomials and other continuous functions you can often just plug in x = c, which is called direct substitution.
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When direct substitution gives 0/0, you have to rewrite the expression before you can find the limit. Common moves are factoring and canceling, multiplying by a conjugate to deal with a square root, combining fractions and using trig identities.
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Much of the skill is choosing the right tool. Try direct substitution first; if you get 0/0, look at the form of the expression to decide whether to factor, use a conjugate, apply an identity or try something else.
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If g(x) ≤ f(x) ≤ h(x) near c, and g and h both have limit L at c, then f is trapped and its limit must be L too. It is the classic way to show that lim (x→0) sin(x)/x = 1 and to find limits like x²·sin(1/x) as x→0.
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The same limit can show up in a graph, a table, an equation or a sentence. You should be able to move between these and check that they all tell the same story.
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A graph can break in three main ways: a removable discontinuity (a hole), a jump discontinuity (the left and right limits are different numbers) and an infinite discontinuity (a vertical asymptote). The type depends on how the limits behave at that spot.
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f is continuous at x = c when three things are true: f(c) exists, lim (x→c) f(x) exists, and the two are equal. Checking all three conditions is how you justify continuity on the AP exam.
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To be continuous on an interval, a function can't break anywhere inside it. Polynomials, rational, exponential, log and trig functions are continuous wherever they are defined, so you mostly check the trouble spots, like zeros of a denominator or where a piecewise rule switches.
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If the limit at x = c exists but f(c) is missing or wrong, you can redefine f at that single point to make it continuous. For a piecewise function, you can pick a constant so the pieces meet with no gap.
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When f(x) grows without bound as x approaches c, you write the limit as ∞ or −∞, and the line x = c is a vertical asymptote. Check the sign of f on each side of c to see whether the graph shoots up or down.
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A limit as x→∞ or x→−∞ describes a function's end behavior; if it equals a number L, the line y = L is a horizontal asymptote. For rational functions, compare the highest powers on the top and bottom to find it quickly.
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If f is continuous on [a, b], then f hits every value between f(a) and f(b) somewhere in that interval. You must confirm continuity before using it, and it tells you a value exists without telling you where.
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