Line is a vertical asymptote for a function, f, if as x approaches 0, f(x) increases or decreases withoutĪsymptote occurs where the function is undefined. Values would tend toward negative infinity. Similarly, if x gets closer and closer to 0 from the negative side, the output Graph of a function and its vertical asymptote(s) do not meet. The word “asymptote” comes from Greek roots Line is what mathematicians call a vertical asymptote. X-coordinates gets very close to 0, the y-coordinates increase without bound Is approaching 0 from the left or negativeĪre getting very large in the positive direction. Values of x can approach 0 from either the right side (where the numbersĪre positive) or from the left side (where the numbers are negative). Since f is undefined at 0, we consider values of x very close, but not equal, to 0. Now, we consider the output values when x gets close to zero. You might convince yourself of this fact by The line is the horizontal asymptote of the graph ofįor any n. Thus, the line is the horizontal asymptote of the graph. Might notice that the same phenomenon occurs for the function g given X decreases without bound, “x approaches negative infinity ” (the behavior on the far left and far right side of the graph) determinesĪsymptote for a function f, if, as the input, x, increases or decreases withoutīound, the output, approaches 0. We say that the line is the horizontal asymptote of You should notice that these points are also getting very close to the Then and we say “as x approaches negative infinity, then approaches 0.” Negative, the output values would also be negative, but also getting close to Points are getting very close to the x-axis as. Pieces, those pieces get smaller and smaller.) In symbols, we write: as ,Īnd we say “as x approaches infinity, approaches 0.” Reciprocal of a very large number is a very small number. We see that as x gets very large, the outputs get close to 0, since the Stretch or shrink of a reciprocal functionĮxercise, you created various tables of values for different functions. Of a reciprocal function through the x-axis Accurately graph by hand the graph of the common reciprocal functions
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