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.
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?
\( \displaystyle \lim_{x \to a} [k \cdot f(x)] \) | \( k = 2 \), \(f(x) = x \) |
\( k = 3 \), \(f(x) = x^2 \) |
\( k = \pi \), \(f(x) = x^2 \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} 2x = 0 \) | \( \displaystyle \lim_{x \to 0} 3x^2 = 0 \) | \( \displaystyle \lim_{x \to 0} \pi x^2 = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} 2x = 2 \) | \( \displaystyle \lim_{x \to 1} 3x^2 = 3 \) | \( \displaystyle \lim_{x \to 1} \pi x^2 = \pi \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} 2x = 200 \) | \( \displaystyle \lim_{x \to 100} 3x^2 = 30000 \) | \( \displaystyle \lim_{x \to 100} \pi x^2 = 10000 \pi \) |
\( \displaystyle k \cdot \lim_{x \to a} f(x) \) | \( k = 2 \) \( f(x) = x \) |
\( k = 3 \), \(f(x) = x^2 \) |
\( k = \pi \), \(f(x) = x^2 \) |
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\( a = 0 \) | \( \displaystyle 2 \cdot \lim_{x \to 0} x = 0 \) | \( \displaystyle 3 \cdot \lim_{x \to 0} x^2 = 0 \) | \( \displaystyle \pi \cdot \lim_{x \to 0} x^2 = 0 \) |
\( a = 1 \) | \( \displaystyle 2 \cdot \lim_{x \to 1} x = 2 \) | \( \displaystyle 3 \cdot \lim_{x \to 1} x^2 = 3 \) | \( \displaystyle \pi \cdot \lim_{x \to 1} x^2 = \pi \) |
\( a = 100 \) | \( \displaystyle 2 \cdot \lim_{x \to 100} x = 200 \) | \( \displaystyle 3 \cdot \lim_{x \to 100} x^2 = 30000 \) | \( \displaystyle \pi \cdot \lim_{x \to 100} x^2 = 10000 \pi \) |
\( \displaystyle \lim_{x \to a} [f(x) + g(x) ] \) | \( f(x) = x \), \(g(x) = x^2 \) |
\( f(x) = 2x \), \(g(x) = x^2 \) |
\( f(x) = x^3 \), \(g(x) = x^2 \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} [x + x^2] = 0 \) | \( \displaystyle \lim_{x \to 0} [2x + x^2] = 0 \) | \( \displaystyle \lim_{x \to 0} [x^3 + x^2] = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} [x + x^2] = 2 \) | \( \displaystyle \lim_{x \to 1} [2x + x^2] = 3 \) | \( \displaystyle \lim_{x \to 1} [x^3 + x^2] = 2 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} [x + x^2] = 10100 \) | \( \displaystyle \lim_{x \to 100} [2x + x^2] = 10200 \) | \( \displaystyle \lim_{x \to 100} [x^3 + x^2] = 1010000 \) |
\( \displaystyle \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \) | \( f(x) = x \), \(g(x) = x^2 \) |
\( f(x) = 2x \), \(g(x) = x^2 \) |
\( f(x) = x^3 \), \(g(x) = x^2 \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} x + \lim_{x \to 0} x^2 = 0 \) | \( \displaystyle \lim_{x \to 0} 2x + \lim_{x \to 0} x^2 = 0 \) | \( \displaystyle \lim_{x \to 0} x^3 + \lim_{x \to 0} x^2 = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} x + \lim_{x \to 1} x^2 = 2 \) | \( \displaystyle \lim_{x \to 1} 2x + \lim_{x \to 1} x^2 = 3 \) | \( \displaystyle \lim_{x \to 1} x^3 + \lim_{x \to 1} x^2 = 2 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} x + \lim_{x \to 100} x^2 = 10100 \) | \( \displaystyle \lim_{x \to 100} 2x + \lim_{x \to 100} x^2 = 10200 \) | \( \displaystyle \lim_{x \to 100} x^3 + \lim_{x \to 100} x^2 = 1010000 \) |
\( \displaystyle \lim_{x \to a} [f(x) \cdot g(x)] \) | \( f(x) = x \), \(g(x) = x \) |
\( f(x) = x^2 \), \(g(x) = 2x \) |
\( f(x) = x^2 \), \(g(x) = \cos(x) \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} [x \cdot x] = 0 \) | \( \displaystyle \lim_{x \to 0} [x^2 \cdot 2x] = 0 \) | \( \displaystyle \lim_{x \to 0} [x^2 \cdot \cos(x)] = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} [x \cdot x] = 1 \) | \( \displaystyle \lim_{x \to 1} [x^2 \cdot 2x] = 2 \) | \( \displaystyle \lim_{x \to 1} [x^2 \cdot \cos(x)] ~= 0.54 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} [x \cdot x] = 10000 \) | \( \displaystyle \lim_{x \to 100} [x^2 \cdot 2x] = 2000000 \) | \( \displaystyle \lim_{x \to 100} [x^2 \cdot \cos(x)] ~= 8623.19 \) |
\( \displaystyle \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \) | \( f(x) = x \), \(g(x) = x \) |
\( f(x) = x^2 \), \(g(x) = 2x \) |
\( f(x) = x^2 \), \(g(x) = \cos(x) \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} x \cdot \lim_{x \to 0} x = 0 \) | \( \displaystyle \lim_{x \to 0} x^2 \cdot \lim_{x \to 0} 2x = 0 \) | \( \displaystyle \lim_{x \to 0} x^2 \cdot \lim_{x \to 0} \cos(x) = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} x \cdot \lim_{x \to 1} x = 1 \) | \( \displaystyle \lim_{x \to 1} x^2 \cdot \lim_{x \to 1} 2x = 2 \) | \( \displaystyle \lim_{x \to 1} x^2 \cdot \lim_{x \to 1} \cos(x) ~= 0.54 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} x \cdot \lim_{x \to 100} x = 10000 \) | \( \displaystyle \lim_{x \to 100} x^2 \cdot \lim_{x \to 100} 2x = 2000000 \) | \( \displaystyle \lim_{x \to 100} x^2 \cdot \lim_{x \to 100} \cos(x) ~= 8623.19 \) |
\( \displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} \) | \( f(x) = x \), \(g(x) = \sin(x) \) |
\( f(x) = x^2 \), \(g(x) = \sin(x) \) |
\( f(x) = x^2 \), \(g(x) = x \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} \big[ \frac{x}{\sin(x)} \big] = 1 \) | \( \displaystyle \lim_{x \to 0} \big[ \frac{x^2}{\sin(x)} \big] = 0 \) | \( \displaystyle \lim_{x \to 0} \big[ \frac{x^2}{x} \big] = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} \big[ \frac{x}{\sin(x)} \big] ~= 1.188 \) | \( \displaystyle \lim_{x \to 1} \big[ \frac{x^2}{\sin(x)} \big] ~= 1.188 \) | \( \displaystyle \lim_{x \to 1} \big[ \frac{x^2}{x} \big] = 1 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} \big[ \frac{x}{\sin(x)} \big] ~= -197.486 \) | \( \displaystyle \lim_{x \to 100} \big[ \frac{x^2}{\sin(x)} \big] ~= -19748.6 \) | \( \displaystyle \lim_{x \to 100} \big[ \frac{x^2}{x} \big] = 100 \) |
\( \frac{\displaystyle \lim_{x \to a} f(x)}{\displaystyle \lim_{x \to a} g(x)} \) | \( f(x) = x \), \(g(x) = \sin(x) \) |
\( f(x) = x^2 \), \(g(x) = \sin(x) \) |
\( f(x) = x^2 \), \(g(x) = x \) |
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\( a = 0 \) | \( \frac{\displaystyle \lim_{x \to 0} x}{\displaystyle \lim_{x \to 0} \sin(x)} = \text{undefined} \) | \( \frac{\displaystyle \lim_{x \to 0} x^2}{\displaystyle \lim_{x \to 0} \sin(x)} = \text{undefined} \) | \( \frac{\displaystyle \lim_{x \to 0} x^2}{\displaystyle \lim_{x \to 0} x} = \text{undefined} \) |
\( a = 1 \) | \( \frac{\displaystyle \lim_{x \to 1} x}{\displaystyle \lim_{x \to 1} \sin(x)} ~= 1.188 \) | \( \frac{\displaystyle \lim_{x \to 1} x^2}{\displaystyle \lim_{x \to 1} \sin(x)} ~= 1.188 \) | \( \frac{\displaystyle \lim_{x \to 1} x^2}{\displaystyle \lim_{x \to 1} x} = 1 \) |
\( a = 100 \) | \( \frac{\displaystyle \lim_{x \to 100} x}{\displaystyle \lim_{x \to 100} \sin(x)} = -197.486 \) | \( \frac{\displaystyle \lim_{x \to 100} x^2}{\displaystyle \lim_{x \to 100} \sin(x)} ~= -19748.6 \) | \( \frac{\displaystyle \lim_{x \to 100} x^2}{\displaystyle \lim_{x \to 100} x} = 100 \) |
\( \displaystyle \lim_{x \to a} [f(x) - g(x)] \) | \( f(x) = x \), \(g(x) = 5 \) |
\( f(x) = 2x \), \(g(x) = x^2 \) |
\( f(x) = x^3 \), \(g(x) = x^2 \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} [x - 5] = -5 \) | \( \displaystyle \lim_{x \to 0} [2x - x^2] = 0 \) | \( \displaystyle \lim_{x \to 0} [x^3 - x^2] = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} [x - 5] = -4 \) | \( \displaystyle \lim_{x \to 1} [2x - x^2] = 1 \) | \( \displaystyle \lim_{x \to 1} [x^3 - x^2] = 0 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} [x - 5] = 95 \) | \( \displaystyle \lim_{x \to 100} [2x - x^2] = -9800 \) | \( \displaystyle \lim_{x \to 100} [x^3 - x^2] = 990000 \) |
\( \displaystyle \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \) | \( f(x) = x \), \(g(x) = 5 \) |
\( f(x) = 2x \), \(g(x) = x^2 \) |
\( f(x) = x^3 \), \(g(x) = x^2 \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} x - \lim_{x \to 0} 5 = -5 \) | \( \displaystyle \lim_{x \to 0} 2x - \lim_{x \to 0} x^2 = 0 \) | \( \displaystyle \lim_{x \to 0} x^3 - \lim_{x \to 0} x^2 = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} x - \lim_{x \to 1} 5 = -4 \) | \( \displaystyle \lim_{x \to 1} 2x - \lim_{x \to 1} x^2 = 1 \) | \( \displaystyle \lim_{x \to 1} x^3 - \lim_{x \to 1} x^2 = 0 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} x - \lim_{x \to 100} 5 = 95 \) | \( \displaystyle \lim_{x \to 100} 2x - \lim_{x \to 100} x^2 = -9800 \) | \( \displaystyle \lim_{x \to 100} x^3 - \lim_{x \to 100} x^2 = 990000 \) |
\( \displaystyle \lim_{x \to a} [f(x)]^c \) | \( f(x) = x \), \( c = 3 \) |
\( f(x) = 2x \), \( c = 3 \) |
\( f(x) = x^2 \), \( c = 5 \) |
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\( a = 0 \) | \( \displaystyle \lim_{x \to 0} [x^3] = 0 \) | \( \displaystyle \lim_{x \to 0} [(2x)^3] = 0 \) | \( \displaystyle \lim_{x \to 0} [(x^2)^5] = 0 \) |
\( a = 1 \) | \( \displaystyle \lim_{x \to 1} [x^3] = 1 \) | \( \displaystyle \lim_{x \to 1} [(2x)^3] = 8 \) | \( \displaystyle \lim_{x \to 1} [(x^2)^5] = 1 \) |
\( a = 100 \) | \( \displaystyle \lim_{x \to 100} [x^3] = 1000000 \) | \( \displaystyle \lim_{x \to 100} [(2x)^3] = 8000000 \) | \( \displaystyle \lim_{x \to 100} [(x^2)^5] = 10^{20} \) |
\( \displaystyle \big[ \lim_{x \to a} f(x) \big]^c \) | \( \displaystyle f(x) = x \), \( c = 3 \) |
\( \displaystyle f(x) = 2x \), \( c = 3 \) |
\( \displaystyle f(x) = x^2 \), \( c = 5 \) |
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\( a = 0 \) | \( \displaystyle \big[ \lim_{x \to 0} x \big]^3 = 0 \) | \( \displaystyle \big[ \lim_{x \to 0} 2x \big]^3 = 0 \) | \( \displaystyle \big[ \lim_{x \to 0} x^2 \big]^5 = 0 \) |
\( a = 1 \) | \( \displaystyle \big[ \lim_{x \to 1} x \big]^3 = 1 \) | \( \displaystyle \big[ \lim_{x \to 1} 2x \big]^3 = 8 \) | \( \displaystyle \big[ \lim_{x \to 1} x^2 \big]^5 = 1 \) |
\( a = 100 \) | \( \displaystyle \big[ \lim_{x \to 100} x \big]^3 = 1000000 \) | \( \displaystyle \big[ \lim_{x \to 100} 2x \big]^3 = 8000000 \) | \( \displaystyle \big[ \lim_{x \to 100} x^2 \big]^5 = 10^{20} \) |