l'hopital presentation notes - google docs
Post on 19-Feb-2017
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L’Hopital’s Rule L’Hopital’s Rule : If f & g are differentiable, f(a) = g(a) = 0, and g’(a) ≠ 0, then,
= limx a→
f (x)g(x)
f (a)′g (a)′
Identify the limit: limx 0→
xe −12x
If you plug in 0 for x, you get , which is undefined.0
0
So you can substitute values close to 0 to get an approximate idea. However, you can use local linearity to calculate the limit. f(x) = e 2x 1 g(x) = x
= f (x)g(x) x
e −12x
= → = 2f (x)′g (x)′ 1
2e 2x
limx 0 →
12e 2x
Practice Problems
1) Find the limit using L’Hopital’s rule if it applies: 1/4limx 2→
(x−2)(x −4) 2
2) Find the limit using L’Hopital’s rule if it applies: 0in(x) e limx 0→
s / x
3) Find the limit using L’Hopital’s rule if it applies: 0n(x) x limx ∞→
l /
4) Find the limit using L’Hopital’s rule if it applies: , a ≠ 0 1/3a^⅔( ) (x ) limx a→
√3 x −√3 a / − a
5) Does L’Hopital’s rule apply? Yes (e limx ∞→
x/ x ∞) ∞/
6) Does L’Hopital’s rule apply? No(1 t 2 t ) limt ∞→
/ − / 2
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