← Calculus I

Unit 2: Limits

Where calculus starts: the tangent and area problems, limits pinned down numerically and with epsilon and delta, continuity proved rather than assumed.

18 topics · 71 questions

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2.1 The Two Problems That Invented Calculus

  1. How it worksHow does the tangent problem and the area problem both get solved by the same limiting idea?
  2. Going furtherWhat happens to a secant line's slope as its second point slides toward the first?
  3. Common mistakeWhy isn't average speed over a whole trip the same question as speed at one instant?
Your turn1 of 2
A particle's position is 4t2 feet at time t seconds. Find its average velocity, in feet per second, from t = 2 to t = 7 seconds.

2.2 Estimating a Limit from a Table of Values

  1. How it worksHow do you build a table of inputs crowding in from both sides to guess a limit?
  2. Worked exampleFind the limit of (sin x)/x as x approaches 0 by evaluating it at x = 0.1, 0.01, and 0.001.
  3. Common mistakeWhy isn't plugging the target value directly into the function the same as finding its limit there?
Your turn1 of 2
Find the limit of sin(4x)/x as x approaches 0.

2.3 Estimating a Limit from a Graph

  1. How it worksHow do you trace a graph from the left and from the right to read off a limit?
  2. Worked exampleFind the limit of a function from its graph at a point where there is an open circle above the curve.
  3. Common mistakeWhy isn't the value of a function at a point always the same as the limit of the function at that point?
  4. Going furtherWhat happens to the limit at a point where a graph has a removable hole versus a jump?
Your turn1 of 2
A function is defined by f(x) = x + 4 for all x except x = 5, where instead f(5) = 3 (a filled dot below an open circle on the graph). Find the limit of f(x) as x approaches 5.

2.4 One-Sided Limits

  1. How it worksHow do you evaluate a limit by approaching a point from only the left or only the right?
  2. Common mistakeWhy doesn't a two-sided limit exist unless both one-sided limits agree?
  3. Worked exampleFind the left-hand and right-hand limits of f(x) = |x|/x as x approaches 0.
  4. Worked exampleFind the left-hand and right-hand limits of the greatest integer function floor(x) at x = 3.
  5. Going furtherWhat happens to a one-sided limit when the function is only defined on one side of the point to begin with?
Your turn1 of 2
Find the limit of f(x) = 4*|x|/x as x approaches 0 from the RIGHT (x → 0+).

2.4 When a Limit Fails to Exist

  1. How it worksHow do a jump discontinuity, an oscillation, and an unbounded spike each break a limit differently?
  2. Worked exampleShow why the limit of sin(1/x) as x approaches 0 fails to exist by tracking the oscillation.
  3. Common mistakeWhy isn't a limit failing to exist the same situation as a limit equal to infinity?
  4. Going furtherWhat happens to a limit when a function oscillates between two values but the oscillations shrink as x approaches the point?
Your turn1 of 2
Consider f(x) = 3 for x < 0 and f(x) = 8 for x ≥ 0, where 3 is not equal to 8. What can you say about the limit of f(x) as x approaches 0?

2.6 The Limit Laws, Proved

  1. How it worksHow does the definition of a limit let you prove the sum law and the product law from scratch?
  2. Common mistakeWhy isn't the limit of a quotient always the quotient of the limits?
  3. Going furtherWhat changes in the limit laws when the limit of the denominator is zero?
  4. Worked exampleProve that the limit of f(x) + g(x) equals the sum of the individual limits, using the epsilon-delta definition.
  5. Worked exampleProve that the limit of a constant times f(x) equals that constant times the limit of f(x), using the epsilon-delta definition.
Your turn1 of 2
Given that the limit of f(x) as x approaches a is 7, and the limit of g(x) as x approaches a is 7, find the limit of 5·f(x) - 2·g(x) as x approaches a.

2.7 Finding Limits by Factoring and Cancelling

  1. How it worksHow does factoring a numerator and denominator expose a common factor you can cancel before taking the limit?
  2. Common mistakeWhy isn't canceling (x - 3) from top and bottom the same as saying x equals 3?
  3. Worked exampleFind the limit of (x3 - 8)/(x - 2) as x approaches 2 by factoring.
  4. Going furtherWhat happens when factoring the numerator and denominator leaves no common factor to cancel at all?
  5. Worked exampleFind the limit of (x2 - 9)/(x - 3) as x approaches 3.
Your turn1 of 2
Find the limit of (x3 - 53)/(x - 5) as x approaches 5.

2.8 Limits Involving Radicals

  1. How it worksHow does multiplying by a conjugate turn a radical expression into one you can cancel?
  2. Going furtherWhat happens to the rationalizing technique when the radical is in the numerator instead of the denominator?
  3. Worked exampleFind the limit of (square root of x - 3)/(x - 9) as x approaches 9 by rationalizing.
Your turn1 of 2
Find the limit of (square root of x - square root of 7)/(x - 7) as x approaches 7.

2.9 The Squeeze Theorem

  1. How it worksHow do two bounding functions squeezing toward the same value force the function between them to match?
  2. Common mistakeWhy doesn't the squeeze theorem work if the two bounding functions approach different values?
  3. Worked exampleFind the limit of x2·sin(1/x) as x approaches 0 by squeezing it between -x2 and x2.
Your turn1 of 2
Find the limit of x2·sin(4/x) + 6 as x approaches 0, using the Squeeze Theorem on the oscillating term.

2.10 The Special Trigonometric Limit sin(x)/x

  1. How it worksHow does comparing the areas of a triangle, a sector, and another triangle squeeze sin(x)/x toward 1?
  2. Worked exampleFind the limit of sin(5x)/x as x approaches 0 using the special trig limit.
  3. Common mistakeWhy isn't the limit of (1 - cos x)/x also equal to 1?
Your turn1 of 2
Find the limit of sin(4x)/sin(4x) as x approaches 0.

2.11 Infinite Limits and Vertical Asymptotes

  1. How it worksHow does the sign of the numerator and denominator near a zero decide whether a function shoots up or down?
  2. Worked exampleFind the vertical asymptotes of f(x) = (x + 1)/(x2 - 4) and the sign of the limit on each side.
  3. Going furtherWhat happens to the sign of an infinite limit when the same vertical asymptote is approached from the left instead of the right?
  4. Common mistakeWhy isn't a limit of infinity actually a real number the function equals at that point?
Your turn1 of 2
For f(x) = (4x + 1)/(x - 4)2, where 4 > 0 and 4 > 0, what happens to f(x) as x approaches 4 from either side?

2.12 Limits at Infinity and Horizontal Asymptotes

  1. How it worksHow do you divide every term by the highest power of x to find a limit as x goes to infinity?
  2. Worked exampleFind the limit of (3x2 + 1)/(2x2 - 5x) as x approaches infinity.
  3. Going furtherWhat happens to the limit at infinity of a rational function when the denominator's degree is larger?
  4. Common mistakeWhy doesn't a horizontal asymptote mean the function can never cross that height?
Your turn1 of 2
Find the limit of (6x2 + 1)/(6x2 - 8x) as x approaches infinity.

1.10 Exploring Types of Discontinuities

  1. How it worksHow do a removable discontinuity, a jump discontinuity, and an infinite discontinuity each break continuity in their own way?
  2. Worked exampleFind the type of discontinuity in f(x) = (x2 - 4)/(x - 2) at x = 2.
  3. Worked exampleFind the type of discontinuity in a step function that jumps from 2 to 5 at x = 3.
  4. Common mistakeWhy isn't every discontinuity in a graph the result of an undefined value at that exact point?
Your turn1 of 2
For f(x) = (x2 - 72)/(x - 7), find the height of the removable hole at x = 7.

1.13 Removing a Discontinuity by Redefining a Function at a Point

  1. How it worksHow do you find the exact value to assign at a hole so the redefined function becomes continuous there?
  2. Worked exampleFind the value to assign f(x) = (x2 - 1)/(x - 1) at x = 1 so that f becomes continuous everywhere.
  3. Common mistakeWhy doesn't redefining a function at one point ever fix a jump discontinuity?
Your turn1 of 2
f(x) = (x - 6)(x + 6)/(x - 6) for x not equal to 6. Find the value to assign f(6) so that f becomes continuous at x = 6.

2.13 Continuity at a Point, Defined Precisely

  1. How it worksHow do a function's value, its limit, and those two matching together define continuity at a point?
  2. Worked exampleShow whether f(x) defined as x + 1 for x not equal to 3, and f(3) = 3, is continuous at x = 3.
  3. Worked exampleShow why f(x) = (x2 - 1)/(x - 1) is not continuous at x = 1 even though its limit exists there.
  4. Common mistakeWhy isn't a function continuous everywhere its formula is defined without any gaps written in?
  5. Going furtherWhat happens to continuity at a point when the function's value and its limit both exist but don't match?
Your turn1 of 2
A function is defined by f(x) = x + 4 for x not equal to 4, and f(4) = 8. Find f(4) minus the limit of f(x) as x approaches 4.

2.14 Continuity on an Interval and the Intermediate Value Theorem

  1. How it worksHow does the Intermediate Value Theorem guarantee a solution exists without ever finding it?
  2. Going furtherWhat happens to the Intermediate Value Theorem's guarantee when the function is only continuous on part of the interval?
  3. Worked exampleProve that f(x) = x3 - x - 1 has a root between x = 1 and x = 2 using the Intermediate Value Theorem.
  4. Worked exampleShow that cos(x) = x has a solution between x = 0 and x = 1 using the Intermediate Value Theorem.
  5. Common mistakeWhy doesn't the Intermediate Value Theorem apply to a function with a break in the middle of the interval?
Your turn1 of 2
For f(x) = x3 - 3x - 3, compute f(0)·f(3).

2.15 The Precise Epsilon-Delta Definition of a Limit

  1. How it worksHow do a shrinking band around the limit and a shrinking band around the input have to trap each other?
  2. Common mistakeWhy isn't epsilon allowed to depend on delta in the definition of a limit?
  3. Worked exampleWatch how choosing epsilon = 0.1 for the limit of 2x as x approaches 3 determines a matching value of delta.
  4. Going furtherWatch how choosing a smaller epsilon forces a smaller delta to keep the function trapped inside the band.
Your turn1 of 2
For the limit of 6x as x approaches 6 (with 6 > 0), find the largest delta, as a function of epsilon, that guarantees |6x - 6·6| < epsilon whenever 0 < |x - 6| < delta.

2.16 Proving a Limit with Epsilon and Delta

  1. How it worksHow do you work backward from |f(x) - L| < epsilon to find a delta that makes it true?
  2. Worked exampleProve that the limit of x2 as x approaches 2 is 4 using the epsilon-delta definition.
  3. Worked exampleProve that the limit of 2x + 1 as x approaches 3 is 7 using the epsilon-delta definition.
  4. Common mistakeWhy isn't finding one delta that works for one epsilon enough to finish an epsilon-delta proof?
Your turn1 of 2
When proving the limit of x2 as x approaches 6 (with 6 > 0) using epsilon-delta, we restrict delta ≤ 1 so that x stays within 1 of a. Under this restriction, find the standard upper bound used for |x + a|.