Single Variable Calculus

Differentiation

01The Derivative as the Slope of the Tangent Line 02Secant Lines and the Limit Definition of the Derivative 03Differentiating 1/x Straight From the Definition 04Tangents to 1/x and the Triangle of Area 2 05Newton and Leibniz Notation, and the Power Rule Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01The Derivative as an Instantaneous Rate of Change 02Rates of Change Without Time: Gradient and Sensitivity 03Easy Limits, 0/0, and One-Sided Limits 04Continuity, Jumps, and Removable Discontinuities 05Infinite Discontinuities: 1/x and Its Derivative 06Differentiable Implies Continuous Problem set0/10 Problem set 20/10 Practice∞

01Two Kinds of Formula, and the Derivative of Sine 02The Derivative of Cosine, and Two Limits at Zero 03Bow and Bowstring: Proving the Two Trig Limits 04Why the Trig Derivatives Hold Only in Radians 05A Geometric Proof: Sine as a Height on the Circle 06Finishing the Proof, and the Product and Quotient Rules Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

01Proving the Product Rule: Change One Factor at a Time 02The Quotient Rule, and the Power Rule for Negative Exponents 03The Chain Rule: Differentiating a Function of a Function 04Higher Derivatives: The Sine Cycle and Three Notations 05The nth Derivative of xⁿ Is n Factorial, by Induction Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

01The Power Rule for Rational Exponents 02The Slope of a Circle, Two Ways 03Implicit Differentiation of a Quartic Curve 04Inverse Functions and the Reflection Across y = x 05Derivatives of the Arctangent and Arcsine Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Easy Limits, 0/0, and One-Sided Limits

Why can some limits be found by plugging in, while every derivative arrives as 0/0 and must be rescued by cancellation?

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Your summary note

1. Easy limits by substitution
  Define an easy limit as one evaluated by plugging in the limiting value, and work an example such as $\lim_{x \to 4} \frac{x+3}{x^2+1} = \frac{7}{17}$ through to its answer.
2. Derivatives are never easy limits
  Write the difference quotient $\lim_{x \to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ , show substitution gives $\frac{0}{0}$ , and state that cancellation is always required with $x \neq x_0$ assumed.
3. Left- and right-hand limit notation
  Record that $x \to x_0^+$ means approaching with $x \to x_0^+$ from the right and $x \to x_0^-$ means $x \to x_0^-$ from the left.
4. Two-sided example with different one-sided limits
  For $f(x) = x+1$ when $x>0$ and $-x+2$ when $x<0$ , compute $\lim_{x\to 0^+} f(x) = 1$ and $\lim_{x\to 0^-} f(x) = 2$ , noting the value $f(0)$ is not used.
Attempt 1 of 2