Calculating limits. (a) In Maple, create a text region and enter the following s

Question

Calculating limits. (a) In Maple, create a text region and enter the following sentence: Finding two-sided limits. Insert an execution group (command prompt) after the the text region. (b) In Maple, create the function f(x) = sin(|x|)/x . ( HINT: Use sin(abs(x))/x.) (c) Plot f(x) with the x-axis varying from ?10 to 10. Notice that there seems to be a jump between y = ?1 and y = 1. Fix the plot by including discont = true in the plot command. (d) QUESTION: Considering the graph, GUESS the value of lim (x->0) f(x)? (e) Calculate the left-hand limit lim (x->0?) f(x)? using Maple. (f) Calculate the right-hand limit of f(x) lim(x->0+) f(x)? using Maple. (g) QUESTION: Does lim (x->0) f(x) exist? Explain your answer. (h) Confirm your answer by calculating [> limit(f(x), x=0);

Explanation / Answer

follow this There are a lot of ways to evaluate this limit. If you are in your first calculus course, you've just learned that lim_{t o 0} sin(t) /t=1. Using that, in this case we have that lim_{x o 0} sin(2x)/x= 2 lim_{x o 0} sin(2x) / (2x) = 2 * 1=2. Next semester, you'll learn L'Hospital's Rule. Basically, L'Hospital's rule says that if f(x)/g(x) is an indeterminate form of type 0/0 or infinity/infinity, then lim_{x o a} f(x)/g(x)=lim_{x o a} f'(x)/g'(x). Applying L'Hospital's rule to this limit gives us lim_{x o 0} sin(2x)/x = lim_{x o 0} 2 cos(2x)/1= 2*1/1=2. Another approach would be to divide the Maclaurin series for sin(2x) by x and then evaluate the limit of the sequence as x approaches 0. (The first term of the series Maclaurin series for sin(2x) /x is 2 which shows that the limit is 2.) So, there are lots of ways to compute the limit. If you are in Calculus I, your teacher is expecting to see the first approach.

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