Example 1. 
Using L’Hospital’s rule:
So the original limit is equal to 
.
Example 2.
L’Hospital’s rule is used in Steps 2 and 3.
A smarter way to do it is using 
in the third line. So that it becomes
. L’Hospital’s rule tells us that this limit is equal to 1/2.
Another take:
Here we view the original expression as 
which is a 
indeterminate form. Then we use L’Hospital’s rule twice to obtain the answer 1/2.
Remark. The Bernoulli numbers 
is defined by the following equation:
In particular, 
. From the observation that 
is an even function, we deduce that 
for all 
.
Using the identity
%0A#card=math&code=%5Cfrac%7Bx%7D%7B%5Cmathrm%7Be%7D%5Ex-1%7D%20%3D%20%5Csum_%7Bk%3D0%7D%5E%7B%5Cinfty%7D%20B_k%20%5Cfrac%7Bx%5Ek%7D%7Bk%21%7D.%20%3D%201%20-%20%5Cfrac%7Bx%7D%7B2%7D%20%2B%20x%5E2%20%5Ccdot%20%28%20%5Ccdots%20%29%0A&id=MT1RQ)
we can easily evaluate the original limit, which is 
.
Example 3. Evaluate
Solution: Because 
%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D#card=math&code=%5Cmathrm%7Be%7D%20%3D%20%5Clim_%7Bx%20%5Cto%200%7D%20%281%2Bx%29%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D&id=kf1aU), so the original limit is an indefinite 
form. We compute it is using the L’Hospital rule.
Let 
%20%3D%20(1%2Bx)%5E%7B1%2Fx%7D#card=math&code=y%20%3D%20y%28x%29%20%3D%20%281%2Bx%29%5E%7B1%2Fx%7D&id=Pprmv). Then 
%5Ccdot%20%5Cln(1%2Bx)#card=math&code=%5Cln%20y%20%3D%20%281%2Fx%29%5Ccdot%20%5Cln%281%2Bx%29&id=YYcFv). So
%5Cln(1%2Bx)%7D%7Bx%5E2%7D.%0A#card=math&code=y%27%20%3D%20%5Cfrac%7By%7D%7B1%2Bx%7D%20%5Ccdot%20%5Cfrac%7Bx-%281%2Bx%29%5Cln%281%2Bx%29%7D%7Bx%5E2%7D.%0A&id=UWhRX)
Using L’Hospital’s rule, we get
%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%20-%20%5Cmathrm%7Be%7D%20%20%7D%7Bx%7D%20%26%3D%20%5Clim%7Bx%5Cto%200%7D%20y’%20%5C%5C%5C%5C%0A%09%26%3D%20%5Cfrac%7B%5Clim%7Bx%5Cto%200%7D%20y%7D%7B%5Clim%7Bx%5Cto%200%7D%20(1%2Bx)%7D%20%20%5Ccdot%20%5Clim%7Bx%5Cto%200%7D%20%5Cfrac%7Bx-(1%2Bx)%5Ccdot%20%5Cln(1%2Bx)%7D%7Bx%5E2%7D%5C%5C%0A%09%26%3D%20%5Cmathrm%7Be%7D%20%5Ccdot%20%5Clim%7Bx%5Cto%200%7D%20%5Cfrac%7B1-%5Cln(1%2Bx)%7D%7B2x%7D%20%5C%5C%5C%5C%0A%09%26%3D%20%5Cmathrm%7Be%7D%20%5Ccdot%20%5Clim%7Bx%20%5Cto%200%7D%20%5Cfrac%7B-%5Cfrac%7B1%7D%7B1%2Bx%7D%7D%7B2%7D%20%5C%5C%5C%5C%0A%09%26%3D%20-%5Cfrac%7B%5Cmathrm%7Be%7D%7D%7B2%7D.%0A%09%5Cend%7Baligned%7D%0A#card=math&code=%5Cbegin%7Baligned%7D%0A%5Clim%7Bx%20%5Cto%200%7D%20%5Cfrac%7B%281%2Bx%29%5E%7B%5Cfrac%7B1%7D%7Bx%7D%7D%20-%20%5Cmathrm%7Be%7D%20%20%7D%7Bx%7D%20%26%3D%20%5Clim%7Bx%5Cto%200%7D%20y%27%20%5C%5C%5C%5C%0A%09%26%3D%20%5Cfrac%7B%5Clim%7Bx%5Cto%200%7D%20y%7D%7B%5Clim%7Bx%5Cto%200%7D%20%281%2Bx%29%7D%20%20%5Ccdot%20%5Clim%7Bx%5Cto%200%7D%20%5Cfrac%7Bx-%281%2Bx%29%5Ccdot%20%5Cln%281%2Bx%29%7D%7Bx%5E2%7D%5C%5C%0A%09%26%3D%20%5Cmathrm%7Be%7D%20%5Ccdot%20%5Clim%7Bx%5Cto%200%7D%20%5Cfrac%7B1-%5Cln%281%2Bx%29%7D%7B2x%7D%20%5C%5C%5C%5C%0A%09%26%3D%20%5Cmathrm%7Be%7D%20%5Ccdot%20%5Clim_%7Bx%20%5Cto%200%7D%20%5Cfrac%7B-%5Cfrac%7B1%7D%7B1%2Bx%7D%7D%7B2%7D%20%5C%5C%5C%5C%0A%09%26%3D%20-%5Cfrac%7B%5Cmathrm%7Be%7D%7D%7B2%7D.%0A%09%5Cend%7Baligned%7D%0A&id=SAsdH)
The L'Hospital rule is applied in the computation of %5Ccdot%20%5Cln(1%2Bx)%7D%7Bx%5E2%7D#card=math&code=%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7Bx-%281%2Bx%29%5Ccdot%20%5Cln%281%2Bx%29%7D%7Bx%5E2%7D&id=UgDBx).
Example 4. Evaluate 
We cannot directly use equivalent infinitesimal substitution in the denominator because it is not a product, but a difference. This is again a 
indeterminate form. We use L’Hospital’s rule to compute the limit:
