L'Hospital Intro
Author
Ruth Dover
Title
L'Hospital Intro
Description
An introduction to L'Hospital's Rule to help show and explain why it works.
Category
Educational Materials
Keywords
Calculus, L'Hospital's Rule, Theorem
URL
http://www.notebookarchive.org/2018-12-bktilgw/
DOI
https://notebookarchive.org/2018-12-bktilgw
Date Added
2018-12-25
Date Last Modified
2018-12-25
File Size
13.88 kilobytes
Supplements
Rights
Redistribution rights reserved
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L’Hospital Introduction
Ruth Dover, Mathematics Instructor (Emeritus), Illinois Mathematics and Science Academy (IMSA), Aurora, IL
This is an introduction to L’Hospital’s Rule. We start with two rather unknown functions. The idea is to consider . Note that ƒ(2) = 0 = g(2).
lim
x2
f(x)
g(x)
Enter the two functions, ƒ and g.
f[x_]:=(x-5)Log[x-1];g[x_]:=(x-)Sin[x-2];
3
x
Try some numerical values of the quotient. x 1.9 is shown, but you may edit this to check other values. What would you guess to be the limit?
f[x]
g[x]
Consider the graph of the quotient.
Plot,{x,1.5,2.5}
f[x]
g[x]
Now look at the graphs of the two functions separately. Change the domain if you wish.
Plot[{f[x],g[x]},{x,1.5,2.5}]
Execute the following cell and use the slider to zoom in until both graphs resemble straight lines.
xcenter=2;startdist=5;mypl=Plot[{f[x],g[x]},{x,xcenter-startdist,xcenter+startdist}];plrange=PlotRange[mypl][[2,2]]-PlotRange[mypl][[2,1]];Manipulate[Plot[{f[x],g[x]},{x,xcenter-(1-.99k)startdist,xcenter+(1-.99k)startdist},PlotStyleThick,PlotRange{f[xcenter]-.5(1-.99k)plrange,f[xcenter]+.5(1-.99k)plrange},Epilog{Red,PointSize[Large],Point[{xcenter,f[xcenter]}]}],{k,0,1}]
Find the derivative of each function at x = 2.
f'[2]g'[2]
What is the ratio of these two derivatives?
Write the equations of the tangent lines to each of these functions at x = 2.
Write the equations of the tangent lines to each of these functions at x = 2.
Now find the limit as x 2 of the ratio of the tangent line expressions.
Essentially, the limit of the ratio of the two functions as they approach 0 when x a is the same as the limit of the ratio of the rates at which the two functions approach 0 when x a.
Essentially, the limit of the ratio of the two functions as they approach 0 when x a is the same as the limit of the ratio of the rates at which the two functions approach 0 when x a.
That is, in this case, near x = 2, g(x) is about twice as big as ƒ(x). Since both functions equal 0 at x = 2, then g is approaching 0 twice as fast as ƒ.
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Cite this as: Ruth Dover, "L'Hospital Intro" from the Notebook Archive (2019), https://notebookarchive.org/2018-12-bktilgw
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