Calc1131W7T4 - L'Hopital's rule

Saturday, 21 April 2018

11:24 AM

Untitled picture.png Machine generated alternative text: The second limit appears to have the form "0/0", but the theorem asserts that nevertheless, there is a limit, and we can find it by taking the limit of the quotients of the derivatives. Thus for example suppose we are asked to find Let's check that the conditions of L'Hopital's rule are satisfied. Setting f(x) lim f@) = lim g@) -2 So by L'Hopital's rule: -1-1 lim — 1 and g@) = —1 —1 -2 1 we have This method also works when the functions approach oo or —oo_ For example consider the limit Ifwe let f@) = = ex then lim f (x) = lim — So by L'Hopital's Rule lim
Untitled picture.png Machine generated alternative text: The second limit appears to have the form "0/0", but the theorem asserts that nevertheless, there is a limit, and we can find it by taking the limit of the quotients of the derivatives. Thus for example suppose we are asked to find Let's check that the conditions of L'Hopital's rule are satisfied. Setting f(x) lim f@) = lim g@) -2 So by L'Hopital's rule: -1-1 lim — 1 and g@) = —1 —1 -2 1 we have This method also works when the functions approach oo or —oo_ For example consider the limit Ifwe let f@) = = ex then lim f (x) = lim — So by L'Hopital's Rule lim Untitled picture.png Machine generated alternative text: Evaluate the following limits using L'Hopital's rule: i) lim x_+5 —x2+25 ii) lim -4 Untitled picture.png Machine generated alternative text: We can use L'Hopital's rule twice if necessary, and even more times. For example; let's consider the limit lim Then Let f(x) = (1 + x) — — 3/4 and g@) — (x lim = lim g@) — But it's not clear that the limit lim exists. Let's go deeper and apply L'Hopital's rule to investigate the limit (2). Again we check the conditions lim f' (x) lim g' (x) — (1+X)A-3 Now we can easily evaluate L = lim Peeling back our innermost application of L'Hopital's rule shows that the limit (2) exists and equals L. We finally return to the limit (1) and conclude that it also exists and equals L.
Untitled picture.png Machine generated alternative text: We can use L'Hopital's rule twice if necessary, and even more times. For example; let's consider the limit lim Then Let f(x) = (1 + x) — — 3/4 and g@) — (x lim = lim g@) — But it's not clear that the limit lim exists. Let's go deeper and apply L'Hopital's rule to investigate the limit (2). Again we check the conditions lim f' (x) lim g' (x) — (1+X)A-3 Now we can easily evaluate L = lim Peeling back our innermost application of L'Hopital's rule shows that the limit (2) exists and equals L. We finally return to the limit (1) and conclude that it also exists and equals L. Untitled picture.png Machine generated alternative text: Let's see what happens when the conditions of L'Hopital's rule are not satisfied. For example, consider the limit Let f(x) = sin@) + 2m and g(x) x , then lim = lim g@) • But cos(x) + 2 O lim does not exist. L'Hopital's theorem is powerless to tell us anything about the limit Ll because the conditions of the theorem are not satisfied However; it is possible to calculate Ll using the pinching theorem: 2 Now consider the limit at x We again check the conditions of L'Hopital's rule lim = 2K and lim g(x) 7T are not both zero: L'Hopital's theorem is again powerless to tell us anything about the limit 142 because the conditions of the theorem are not satisfied. However, it is possible to calculate directly: 2 Note: the Maple syntax for sin(ff) is sin (Pi) Untitled picture.png Machine generated alternative text: At this point we have many ways to calculate the limit: —3 Which of the following methods do you prefer? O O O O Pinching theorem L'Hopital's rule divide by the highest power of in the denominator Put a really big value of x into the calculator and see approximately what you get. Wolfram Alpha.
Untitled picture.png Machine generated alternative text: At this point we have many ways to calculate the limit: —3 Which of the following methods do you prefer? O O O O Pinching theorem L'Hopital's rule divide by the highest power of in the denominator Put a really big value of x into the calculator and see approximately what you get. Wolfram Alpha. Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings

 

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