Wolfram|Alpha & Limit Laws

Wolfram|Alpha is a self-described computation knowledge engine. For our purposes, we will use it as a computer algebra system similar to Mathematica or Maple.

Examples

For fun

Tasks

1. Record at least two commands to compute a one-sided limit, e.g., \(\displaystyle \lim_{x \to 0^-} \frac{1}{x} \)


2. Record at least two commands to compute a limit, e.g., \( \displaystyle \lim_{x \to 0} \frac{1}{x^2}\)


3. Complete the tables below. What patterns do you notice?

Constant multiple of a limits
\( \displaystyle k \cdot \lim_{x \to a} f(x) \) \( k = 2 \),
\(f(x) = x \)
\( k = 3 \),
\(f(x) = x^2 \)
\( k = \pi \),
\(f(x) = x^2 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

 

Limit constant multiples
\( \displaystyle \lim_{x \to a} [k \cdot f(x)] \) \( k = 2 \),
\(f(x) = x \)
\( k = 3 \),
\(f(x) = x^2 \)
\( k = \pi \),
\(f(x) = x^2 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

Limit of sums
\( \displaystyle \lim_{x \to a} [f(x) + g(x) ] \) \( f(x) = x \),
\(g(x) = x^2 \)
\( f(x) = 2x \),
\(g(x) = x^2 \)
\( f(x) = x^3 \),
\(g(x) = x^2 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

 

Sum of limits
\( \displaystyle \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \) \( f(x) = x \),
\(g(x) = x^2 \)
\( f(x) = 2x \),
\(g(x) = x^2 \)
\( f(x) = x^3 \),
\(g(x) = x^2 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

Limit of products
\( \displaystyle \lim_{x \to a} [f(x) \cdot g(x)] \) \( f(x) = x \),
\(g(x) = x \)
\( f(x) = x^2 \),
\(g(x) = 2x \)
\( f(x) = x^2 \),
\(g(x) = \cos(x) \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

 

Product of limits
\( \displaystyle \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \) \( f(x) = x \),
\(g(x) = x \)
\( f(x) = x^2 \),
\(g(x) = 2x \)
\( f(x) = x^2 \),
\(g(x) = \cos(x) \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

Limit of division
\( \displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} \) \( f(x) = x \),
\(g(x) = \sin(x) \)
\( f(x) = x^2 \),
\(g(x) = \sin(x) \)
\( f(x) = x^2 \),
\(g(x) = x \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

 

Division of limits
\( \frac{\displaystyle \lim_{x \to a} f(x)}{\displaystyle \lim_{x \to a} g(x)} \) \( f(x) = x \),
\(g(x) = \sin(x) \)
\( f(x) = x^2 \),
\(g(x) = \sin(x) \)
\( f(x) = x^2 \),
\(g(x) = x \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

Limit of subtraction
\( \displaystyle \lim_{x \to a} [f(x) - g(x)] \) \( f(x) = x \),
\(g(x) = 5 \)
\( f(x) = 2x \),
\(g(x) = x^2 \)
\( f(x) = x^3 \),
\(g(x) = x^2 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

 

Subtraction of limits
\( \displaystyle \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \) \( f(x) = x \),
\(g(x) = 5 \)
\( f(x) = 2x \),
\(g(x) = x^2 \)
\( f(x) = x^3 \),
\(g(x) = x^2 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

Limit of powers
\( \displaystyle \lim_{x \to a} [f(x)]^c \) \( f(x) = x \),
\( c = 3 \)
\( f(x) = 2x \),
\( c = 3 \)
\( f(x) = x^2 \),
\( c = 5 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)

 

Powers of limits
\( \displaystyle \big[ \lim_{x \to a} f(x) \big]^c \) \( \displaystyle f(x) = x \),
\( c = 3 \)
\( \displaystyle f(x) = 2x \),
\( c = 3 \)
\( \displaystyle f(x) = x^2 \),
\( c = 5 \)
\( a = 0 \)
\( a = 1 \)
\( a = 100 \)