L’Hospital’s Rule Exploration

Directions: Using the digits 1 to 9, at most one time each, create 3 different expressions such that their graphs contains any 2 of the 3 following criteria:

1) Horizontal Asymptote @ y = some positive rational number
2) Slant Asymptote with a slope such that: 1 < m ≤ 2
3) Two Vertical Asymptotes


Top choices vary as they are limited by bottom choices {bot: 8,1,7}

Is it possible to get a positive and negative asymptote with our restrictions?

Vert Asymptote: First term exponent < 2; Set bottom = Zero Slant Asymptote: First term exponent > 2; Slope set by coefficients of first terms simplified

Horiz. Asymptote: First term exponent = 2: Location by coefficients of first terms simplified

All the above hold true because LARGEST term exponent on bottom is “”2″”


In order left to right: {top: varies – limited by bottom choices — bot: 3,10,1,7}

To get criteria 2 and 3 the top (first term) coefficient must be larger than the bottom
(first term) coefficient… therefore top lead coefficient > 3 in this solution example.

Source: Gregory L. Taylor, Ed.D.

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