Limits problem 06.html

1. Piecewise defined functions, One-sided limits, and Infinite limits.
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1. Piecewise defined functions, One-sided limits, and Infinite limits.

Piecewise defined functions play an important role in mathematics, and are defined in Maple using the piecewise command. Look up piecewise in the Maple help to get some idea of how it works. Let us consider the function defined as follows.

> f:=x->piecewise(x<1,x^2,x>=1,x+3);

Let us plot the function to see what its graph will tell us about the limits from the left and right as x approaches 1. Note the use of the option discont=true in the plot command. This warns Maple that the function may have jumps (discontinuities), which affects how Maple will draw the graph.

> plot(f(x),x=0..2,discont=true);

From the picture, we see that it appears that and . Maple can do one sided limits as well.

> limit(f(x),x=1,left); limit(f(x),x=1,right);

Let us consider a function that has an infinite limit.

> f:=x->x/(x+1);

Let us plot the function near , to see what is happening.

> plot(f(x),x=-2..0,y=-10..10,discont=true);

If you leave out the discont=true option, then Maple appears to draw the vertical asymptote in the plot. This is really a bit of an illusion, because Maple's plot command does not really understand the function well enough to show the vertical asymptotes, and it is really playing a game of connect up the dots to obtain the vertical line in the picture. Notice that we also chose a bounded range for the y coordinates in the display, which makes the picture look better. Experiment with leaving out the y range and the discont=true to see the differences in the plot you obtain.