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The sharp point the graph of has at the origin in Figure 2.1.3(a) is an example of a cusp.
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Consider secant lines through and , the slopes of which are
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The limiting values of these slopes are , the positive value when the limit is taken from the right; the negative, from the left.
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Consequently, the limit of these slopes does not exist, so no unique slope can be assigned at .
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plot(abs(x)^(1/3),x=-1..1,scaling=constrained,tickmarks=[3,2],labels=[x,y]);
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Figure 2.1.3(a) Graph of
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The following calculations verify the conclusion that there is no unique slope at .
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Define the function
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Control-drag
Context Panel: Evaluate and Display Inline
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Limiting slopes of the secant lines
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Expression palette: Limit operator
Limit from the left
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=
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Expression palette: Limit operator
Limit from the right
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=
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If the tangent line is a line whose slope is the limiting value of the slopes of secant lines, then this function has no tangent line at the origin. However, if a more geometric notion of a tangent line is invoked, then it could be said that the -axis is a line tangent to the curve at the origin. Hence, the question about the existence of a tangent line hinges on perspective, either analytic or geometric.
