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    • ▾Calculus 1
      • ▾Limits and continuity
        • •Intro to calc 1
        • •Estimating limits from graphs
        • •Estimating limits from tables
        • •Intro to continuity
        • •Finding and classifying discontinuities
        • •Finding the value that makes the function continuous
      • ▸Derivatives: definition and basic derivative rules
        • •Proving differentiable implies continuous
        • •Derivative of a function from first principles
        • •Finding the tangent or normal line through the given point
        • •Constant rule
        • •Constant multiple rule for derivatives
        • •Sum, difference, and power rules for derivatives
        • •Product rule
        • •Proving the product rule
        • •Quotient rule
        • •Derivatives of trig functions
        • •Proving properties of even and odd functions (calculus 1)
        • •Leibniz's derivative notation
        • •Logarithmic functions
      • ▸Derivatives: composite, implicit, and inverse functions
        • •Derivative at a point from first principles
        • •Derivatives of exponential functions
        • •Derivatives of logarithmic functions
        • ▸Derivatives of hyperbolic and inverse hyperbolic functions
          • •Using the derivatives of the hyperbolic functions
          • •Proving the derivatives of the hyperbolic functions
          • •Proving the derivatives of the inverse hyperbolic functions
        • •Derivatives of inverse trig functions
        • •Derivatives of transcendental functions
        • •Derivatives of trigonometric functions under transformations
        • •Derivative using u-substitution
        • •Determining derivatives from graphs
        • •Higher order derivatives
        • •Chain rule
        • •Derivatives of inverse functions
        • •Choosing the derivative rules to use
      • ▸Applying derivatives to analyze functions
        • •Applications of trigonometric derivatives
        • •Extreme value theorem
        • •First derivative test
        • •Mean value theorem
        • ▸Monotonic intervals and functions
          • •Intro
          • •Proofs
        • •Related rates
        • •Second derivative test
        • •Concavity
        • •Factoring a cubic polynomial with a double root
        • •Graphing using derivatives
        • ▸Special points
          • ▸Finding critical values from graphs
            • •Lesson
            • •Practice
          • •Finding critical values using derivatives
          • ▸Inflection points
            • •Graphically finding points of inflection
            • •Using the second derivative to find points of inflection
          • •Classifying stationary points
          • •Sketching polynomials using stationary points
        • •Bisection method
        • •Newton's method
        • •False position method
        • •L'Hopital's rule
      • ▸Integrals
        • •Basic integration problems
        • ▸Integration by u-substitution
          • ▸Integration by u-substitution (less difficult)
            • •Lesson
            • •Practice
          • •Integration by u-substitution (more difficult)
          • •Integrating exponential functions by u-substitution
        • •Visually determining antiderivative
      • ▸Applications of integrals
        • •Proving the formula for the area of a circle
        • •Area between two curves
        • ▸Solids of revolution
          • •Disc and washer methods (circular cross sections)
          • •Volumes of solids with known cross sections
          • •Shell method
          • •Volume of a sphere (calculus \(2\))
          • •Volume of a cone (calculus 2)
     › Calculus 1 › Limits and continuity

    Estimating limits from graphs

    First, students will learn how to evaluate a limit graphically, when the limit exists.

    Watch these videos:

    • Introduction to Limits by NancyPi
    • Estimating limit values from graphs by Khan Academy

    Next, students will learn to recognize when a functions value is approaching positive or negative infinity, from either side, and what it tells us about the limit at that asymptote.

    Unbounded limits by Khan Academy

    After that, students will practice both skills.

    Estimating limit values from graphs by Khan Academy

    Next, students will learn how to evaluate a one-sided limit graphically, when the limit exists.

    One-sided limits from graphs by Khan Academy

    After that, students will learn to recognize when a functions value is approaching positive or negative infinity, from one side, and what it tells us about the one-sided limit at that asymptote.

    One-sided limits from graphs: asymptote by Khan Academy

    After that, students will practice both skills.

    One-sided limits from graphs

    After that, students will learn how to evaluate more complicated limits graphically.

    Ex 1: Determine Limits from a Graph Using Function Notation by Mathispower4u

    Next, students will learn that a limit only tells us about the values around a point. For any limit, we could construct infinitely many functions which have that limit in common. Students will also learn that we needn't always take the limit at points of discontinuity, or at asymptotes. We also needn't take the limit at integer values, we could take the limit of \(f(x)\) as \(x\) approaches \(\pi,\) for example.

    Connecting limits and graphical behavior by Khan Academy

    Connecting limits and graphical behavior by Khan Academy

    Conclude by giving your students these challenges:

    • Ex 2: Determine Limits from a Graph Using Function Notation (Challenging) by Mathispower4u
    • Ex 2: Determine Limits from a Graph Using Function Notation (Challenging) by Mathispower4u
    • Gabriel's Problem Posters by NRICH
    • 2012 AMC 8, Problem 20
    • A Complex Compound Function

    Additional lessons and practice problems