WolframAlpha & Limit Laws

WolframAlpha 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

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

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

Constant multiplier

$$k = 2$$, $$f(x) = x$$ $$k = 3$$, $$f(x) = x^2$$ $$k = \pi$$, $$f(x) = x^2$$ $$\displaystyle \lim_{x \to a} [k \cdot f(x)]$$ $$\displaystyle \lim_{x \to 0} 2x = 0$$ $$\displaystyle \lim_{x \to 0} 3x^2 = 0$$ $$\displaystyle \lim_{x \to 0} \pi x^2 = 0$$ $$\displaystyle \lim_{x \to 1} 2x = 2$$ $$\displaystyle \lim_{x \to 1} 3x^2 = 3$$ $$\displaystyle \lim_{x \to 1} \pi x^2 = \pi$$ $$\displaystyle \lim_{x \to 100} 2x = 200$$ $$\displaystyle \lim_{x \to 100} 3x^2 = 30000$$ $$\displaystyle \lim_{x \to 100} \pi x^2 = 10000 \pi$$
$$k = 2$$ $$f(x) = x$$ $$k = 3$$, $$f(x) = x^2$$ $$k = \pi$$, $$f(x) = x^2$$ $$\displaystyle k \cdot \lim_{x \to a} f(x)$$ $$\displaystyle 2 \cdot \lim_{x \to 0} x = 0$$ $$\displaystyle 3 \cdot \lim_{x \to 0} x^2 = 0$$ $$\displaystyle \pi \cdot \lim_{x \to 0} x^2 = 0$$ $$\displaystyle 2 \cdot \lim_{x \to 1} x = 2$$ $$\displaystyle 3 \cdot \lim_{x \to 1} x^2 = 3$$ $$\displaystyle \pi \cdot \lim_{x \to 1} x^2 = \pi$$ $$\displaystyle 2 \cdot \lim_{x \to 100} x = 200$$ $$\displaystyle 3 \cdot \lim_{x \to 100} x^2 = 30000$$ $$\displaystyle \pi \cdot \lim_{x \to 100} x^2 = 10000 \pi$$

Sums

$$f(x) = x$$, $$g(x) = x^2$$ $$f(x) = 2x$$, $$g(x) = x^2$$ $$f(x) = x^3$$, $$g(x) = x^2$$ $$\displaystyle \lim_{x \to a} [f(x) + g(x) ]$$ $$\displaystyle \lim_{x \to 0} [x + x^2] = 0$$ $$\displaystyle \lim_{x \to 0} [2x + x^2] = 0$$ $$\displaystyle \lim_{x \to 0} [x^3 + x^2] = 0$$ $$\displaystyle \lim_{x \to 1} [x + x^2] = 2$$ $$\displaystyle \lim_{x \to 1} [2x + x^2] = 3$$ $$\displaystyle \lim_{x \to 1} [x^3 + x^2] = 2$$ $$\displaystyle \lim_{x \to 100} [x + x^2] = 10100$$ $$\displaystyle \lim_{x \to 100} [2x + x^2] = 10200$$ $$\displaystyle \lim_{x \to 100} [x^3 + x^2] = 1010000$$
$$f(x) = x$$, $$g(x) = x^2$$ $$f(x) = 2x$$, $$g(x) = x^2$$ $$f(x) = x^3$$, $$g(x) = x^2$$ $$\displaystyle \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$ $$\displaystyle \lim_{x \to 0} x + \lim_{x \to 0} x^2 = 0$$ $$\displaystyle \lim_{x \to 0} 2x + \lim_{x \to 0} x^2 = 0$$ $$\displaystyle \lim_{x \to 0} x^3 + \lim_{x \to 0} x^2 = 0$$ $$\displaystyle \lim_{x \to 1} x + \lim_{x \to 1} x^2 = 2$$ $$\displaystyle \lim_{x \to 1} 2x + \lim_{x \to 1} x^2 = 3$$ $$\displaystyle \lim_{x \to 1} x^3 + \lim_{x \to 1} x^2 = 2$$ $$\displaystyle \lim_{x \to 100} x + \lim_{x \to 100} x^2 = 10100$$ $$\displaystyle \lim_{x \to 100} 2x + \lim_{x \to 100} x^2 = 10200$$ $$\displaystyle \lim_{x \to 100} x^3 + \lim_{x \to 100} x^2 = 1010000$$

Products

$$f(x) = x$$, $$g(x) = x$$ $$f(x) = x^2$$, $$g(x) = 2x$$ $$f(x) = x^2$$, $$g(x) = \cos(x)$$ $$\displaystyle \lim_{x \to a} [f(x) \cdot g(x)]$$ $$\displaystyle \lim_{x \to 0} [x \cdot x] = 0$$ $$\displaystyle \lim_{x \to 0} [x^2 \cdot 2x] = 0$$ $$\displaystyle \lim_{x \to 0} [x^2 \cdot \cos(x)] = 0$$ $$\displaystyle \lim_{x \to 1} [x \cdot x] = 1$$ $$\displaystyle \lim_{x \to 1} [x^2 \cdot 2x] = 2$$ $$\displaystyle \lim_{x \to 1} [x^2 \cdot \cos(x)] ~= 0.54$$ $$\displaystyle \lim_{x \to 100} [x \cdot x] = 10000$$ $$\displaystyle \lim_{x \to 100} [x^2 \cdot 2x] = 2000000$$ $$\displaystyle \lim_{x \to 100} [x^2 \cdot \cos(x)] ~= 8623.19$$
$$f(x) = x$$, $$g(x) = x$$ $$f(x) = x^2$$, $$g(x) = 2x$$ $$f(x) = x^2$$, $$g(x) = \cos(x)$$ $$\displaystyle \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$$ $$\displaystyle \lim_{x \to 0} x \cdot \lim_{x \to 0} x = 0$$ $$\displaystyle \lim_{x \to 0} x^2 \cdot \lim_{x \to 0} 2x = 0$$ $$\displaystyle \lim_{x \to 0} x^2 \cdot \lim_{x \to 0} \cos(x) = 0$$ $$\displaystyle \lim_{x \to 1} x \cdot \lim_{x \to 1} x = 1$$ $$\displaystyle \lim_{x \to 1} x^2 \cdot \lim_{x \to 1} 2x = 2$$ $$\displaystyle \lim_{x \to 1} x^2 \cdot \lim_{x \to 1} \cos(x) ~= 0.54$$ $$\displaystyle \lim_{x \to 100} x \cdot \lim_{x \to 100} x = 10000$$ $$\displaystyle \lim_{x \to 100} x^2 \cdot \lim_{x \to 100} 2x = 2000000$$ $$\displaystyle \lim_{x \to 100} x^2 \cdot \lim_{x \to 100} \cos(x) ~= 8623.19$$

Quotients

$$f(x) = x$$, $$g(x) = \sin(x)$$ $$f(x) = x^2$$, $$g(x) = \sin(x)$$ $$f(x) = x^2$$, $$g(x) = x$$ $$\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)}$$ $$\displaystyle \lim_{x \to 0} \big[ \frac{x}{\sin(x)} \big] = 1$$ $$\displaystyle \lim_{x \to 0} \big[ \frac{x^2}{\sin(x)} \big] = 0$$ $$\displaystyle \lim_{x \to 0} \big[ \frac{x^2}{x} \big] = 0$$ $$\displaystyle \lim_{x \to 1} \big[ \frac{x}{\sin(x)} \big] ~= 1.188$$ $$\displaystyle \lim_{x \to 1} \big[ \frac{x^2}{\sin(x)} \big] ~= 1.188$$ $$\displaystyle \lim_{x \to 1} \big[ \frac{x^2}{x} \big] = 1$$ $$\displaystyle \lim_{x \to 100} \big[ \frac{x}{\sin(x)} \big] ~= -197.486$$ $$\displaystyle \lim_{x \to 100} \big[ \frac{x^2}{\sin(x)} \big] ~= -19748.6$$ $$\displaystyle \lim_{x \to 100} \big[ \frac{x^2}{x} \big] = 100$$
$$f(x) = x$$, $$g(x) = \sin(x)$$ $$f(x) = x^2$$,$$g(x) = \sin(x)$$ $$f(x) = x^2$$,$$g(x) = x$$ $$\frac{\displaystyle \lim_{x \to a} f(x)}{\displaystyle \lim_{x \to a} g(x)}$$ $$\frac{\displaystyle \lim_{x \to 0} x}{\displaystyle \lim_{x \to 0} \sin(x)} = \text{undefined}$$ $$\frac{\displaystyle \lim_{x \to 0} x^2}{\displaystyle \lim_{x \to 0} \sin(x)} = \text{undefined}$$ $$\frac{\displaystyle \lim_{x \to 0} x^2}{\displaystyle \lim_{x \to 0} x} = \text{undefined}$$ $$\frac{\displaystyle \lim_{x \to 1} x}{\displaystyle \lim_{x \to 1} \sin(x)} ~= 1.188$$ $$\frac{\displaystyle \lim_{x \to 1} x^2}{\displaystyle \lim_{x \to 1} \sin(x)} ~= 1.188$$ $$\frac{\displaystyle \lim_{x \to 1} x^2}{\displaystyle \lim_{x \to 1} x} = 1$$ $$\frac{\displaystyle \lim_{x \to 100} x}{\displaystyle \lim_{x \to 100} \sin(x)} = -197.486$$ $$\frac{\displaystyle \lim_{x \to 100} x^2}{\displaystyle \lim_{x \to 100} \sin(x)} ~= -19748.6$$ $$\frac{\displaystyle \lim_{x \to 100} x^2}{\displaystyle \lim_{x \to 100} x} = 100$$

Subtraction

$$f(x) = x$$, $$g(x) = 5$$ $$f(x) = 2x$$, $$g(x) = x^2$$ $$f(x) = x^3$$, $$g(x) = x^2$$ $$\displaystyle \lim_{x \to a} [f(x) - g(x)]$$ $$\displaystyle \lim_{x \to 0} [x - 5] = -5$$ $$\displaystyle \lim_{x \to 0} [2x - x^2] = 0$$ $$\displaystyle \lim_{x \to 0} [x^3 - x^2] = 0$$ $$\displaystyle \lim_{x \to 1} [x - 5] = -4$$ $$\displaystyle \lim_{x \to 1} [2x - x^2] = 1$$ $$\displaystyle \lim_{x \to 1} [x^3 - x^2] = 0$$ $$\displaystyle \lim_{x \to 100} [x - 5] = 95$$ $$\displaystyle \lim_{x \to 100} [2x - x^2] = -9800$$ $$\displaystyle \lim_{x \to 100} [x^3 - x^2] = 990000$$
$$f(x) = x$$, $$g(x) = 5$$ $$f(x) = 2x$$, $$g(x) = x^2$$ $$f(x) = x^3$$, $$g(x) = x^2$$ $$\displaystyle \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$$ $$\displaystyle \lim_{x \to 0} x - \lim_{x \to 0} 5 = -5$$ $$\displaystyle \lim_{x \to 0} 2x - \lim_{x \to 0} x^2 = 0$$ $$\displaystyle \lim_{x \to 0} x^3 - \lim_{x \to 0} x^2 = 0$$ $$\displaystyle \lim_{x \to 1} x - \lim_{x \to 1} 5 = -4$$ $$\displaystyle \lim_{x \to 1} 2x - \lim_{x \to 1} x^2 = 1$$ $$\displaystyle \lim_{x \to 1} x^3 - \lim_{x \to 1} x^2 = 0$$ $$\displaystyle \lim_{x \to 100} x - \lim_{x \to 100} 5 = 95$$ $$\displaystyle \lim_{x \to 100} 2x - \lim_{x \to 100} x^2 = -9800$$ $$\displaystyle \lim_{x \to 100} x^3 - \lim_{x \to 100} x^2 = 990000$$

Powers

$$f(x) = x$$, $$c = 3$$ $$f(x) = 2x$$, $$c = 3$$ $$f(x) = x^2$$, $$c = 5$$ $$\displaystyle \lim_{x \to a} [f(x)]^c$$ $$\displaystyle \lim_{x \to 0} [x^3] = 0$$ $$\displaystyle \lim_{x \to 0} [(2x)^3] = 0$$ $$\displaystyle \lim_{x \to 0} [(x^2)^5] = 0$$ $$\displaystyle \lim_{x \to 1} [x^3] = 1$$ $$\displaystyle \lim_{x \to 1} [(2x)^3] = 8$$ $$\displaystyle \lim_{x \to 1} [(x^2)^5] = 1$$ $$\displaystyle \lim_{x \to 100} [x^3] = 1000000$$ $$\displaystyle \lim_{x \to 100} [(2x)^3] = 8000000$$ $$\displaystyle \lim_{x \to 100} [(x^2)^5] = 10^{20}$$
$$\displaystyle f(x) = x$$, $$c = 3$$ $$\displaystyle f(x) = 2x$$, $$c = 3$$ $$\displaystyle f(x) = x^2$$, $$c = 5$$ $$\displaystyle \big[ \lim_{x \to a} f(x) \big]^c$$ $$\displaystyle \big[ \lim_{x \to 0} x \big]^3 = 0$$ $$\displaystyle \big[ \lim_{x \to 0} 2x \big]^3 = 0$$ $$\displaystyle \big[ \lim_{x \to 0} x^2 \big]^5 = 0$$ $$\displaystyle \big[ \lim_{x \to 1} x \big]^3 = 1$$ $$\displaystyle \big[ \lim_{x \to 1} 2x \big]^3 = 8$$ $$\displaystyle \big[ \lim_{x \to 1} x^2 \big]^5 = 1$$ $$\displaystyle \big[ \lim_{x \to 100} x \big]^3 = 1000000$$ $$\displaystyle \big[ \lim_{x \to 100} 2x \big]^3 = 8000000$$ $$\displaystyle \big[ \lim_{x \to 100} x^2 \big]^5 = 10^{20}$$