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Understanding Limits: Cheat Sheet

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Essential Concepts

A Preview of Calculus

  • Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. The slope of the tangent line indicates the rate of change of the function, also called the derivative. Calculating a derivative requires finding a limit.
  • Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x-axis. We can approximate the area by dividing it into thin rectangles and summing the areas of these rectangles. This summation leads to the value of a function called the integral. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function.
  • Multivariable calculus enables us to solve problems in three-dimensional space, including determining motion in space and finding volumes of solids.

Introduction to the limit of a function

  • A table of values or graph may be used to estimate a limit.
  • If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist.
  • If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value.
  • We may use limits to describe infinite behavior of a function at a point.

Key Equations

  • Slope of a Secant Line
    msec=f(x)f(a)xa
  • Average Velocity over Interval [a,t]
    vavg=s(t)s(a)ta
  • One-Sided Limits
    limxaf(x)=L
    limxa+f(x)=L
  • Intuitive Definition of the Limit
    limxaf(x)=L

Glossary

infinite limit
A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
intuitive definition of the limit
If all values of the function f(x) approach the real number L as the values of x(a) approach a, f(x) approaches L
one-sided limit
A one-sided limit of a function is a limit taken from either the left or the right
vertical asymptote
A function has a vertical asymptote at x=a if the limit as x approaches a from the right or left is infinite

Study Tips

The Tangent Problem and Differential Calculus

  • Practice sketching secant lines for various functions and intervals.
  • Visualize how secant lines approach the tangent line as points get closer.
  • When working with position functions, relate concepts of average and instantaneous velocity to secant and tangent lines.
  • Practice calculating average velocities over smaller and smaller intervals.

The Area Problem and Integral Calculus

  • Practice approximating areas using different numbers of rectangles. Understand the relationship between smaller rectangles and better approximations.
  • Visualize how the sum of rectangle areas approaches the true area as widths decrease.
  • Practice setting up integrals for various shaped regions.

The Definition of a Limit

  • Practice estimating limits using both tables and graphs.
  • Be aware that a function can be defined at a point but have a different limit there.
  • Look for oscillating behavior when suspecting a limit might not exist.
  • Remember that limits describe behavior near a point, not necessarily at the point.
  • Use basic limit properties to simplify complex limit problems.

One-Sided and Two-Sided Limits

  • When evaluating limits, always consider both left-hand and right-hand approaches.
  • For piecewise functions, use the appropriate piece for each one-sided limit.
  • Create tables of values approaching the point of interest from both sides.
  • Remember that a function can be continuous from one side but not the other.
  • When solving limit problems, explicitly state both one-sided limits before concluding about the two-sided limit.

Infinite Limits

  • Remember that infinite limits are not actual limits but descriptions of behavior.
  • When finding vertical asymptotes, check both sides of the potential asymptote.
  • Use factoring to identify potential vertical asymptotes in rational functions.
  • Distinguish between vertical asymptotes and holes in rational functions.