All GRE Subject Test: Math Resources
Example Questions
Example Question #1 : Euler's Method And L'hopital's Rule
Evaluate:
The limit does not exist.
Let's examine the limit
first.
and
,
so by L'Hospital's Rule,
Since ,
Now, for each , ; therefore,
By the Squeeze Theorem,
and
Example Question #1 : Limits
Evaluate:
The limit does not exist.
Therefore, by L'Hospital's Rule, we can find by taking the derivatives of the expressions in both the numerator and the denominator:
Similarly,
So
But for any , so
Example Question #3 : Euler's Method And L'hopital's Rule
Evaluate:
The limit does not exist.
and
Therefore, by L'Hospital's Rule, we can find by taking the derivatives of the expressions in both the numerator and the denominator:
Similarly,
so
Example Question #2 : Limits
Evaluate:
and
Therefore, by L'Hospital's Rule, we can find by taking the derivatives of the expressions in both the numerator and the denominator:
Example Question #1 : L'hospital's Rule
Solve:
Substitution is invalid. In order to solve , rewrite this as an equation.
Take the natural log of both sides to bring down the exponent.
Since is in indeterminate form, , use the L'Hopital Rule.
L'Hopital Rule is as follows:
This indicates that the right hand side of the equation is zero.
Use the term to eliminate the natural log.
Example Question #1 : L'hospital's Rule
Evaluate the limit using L'Hopital's Rule.
Undefined
L'Hopital's Rule is used to evaluate complicated limits. The rule has you take the derivative of both the numerator and denominator individually to simplify the function. In the given function we take the derivatives the first time and get
.
This still cannot be evaluated properly, so we will take the derivative of both the top and bottom individually again. This time we get
.
Now we have only one x, so we can evaluate when x is infinity. Plug in infinity for x and we get
and .
So we can simplify the function by remembering that any number divided by infinity gives you zero.
Example Question #5 : Euler's Method And L'hopital's Rule
Evaluate the limit using L'Hopital's Rule.
Undefined
L'Hopital's Rule is used to evaluate complicated limits. The rule has you take the derivative of both the numerator and denominator individually to simplify the function. In the given function we take the derivatives the first time and get
.
Since the first set of derivatives eliminates an x term, we can plug in zero for the x term that remains. We do this because the limit approaches zero.
This gives us
.
Example Question #2 : L'hospital's Rule
Evaluate the limit using L'Hopital's Rule.
Undefined
L'Hopital's Rule is used to evaluate complicated limits. The rule has you take the derivative of both the numerator and denominator individually to simplify the function. In the given function we take the derivatives the first time and get
.
This still cannot be evaluated properly, so we will take the derivative of both the top and bottom individually again. This time we get
.
Now we have only one x, so we can evaluate when x is infinity. Plug in infinity for x and we get
.
Example Question #3 : L'hospital's Rule
Calculate the following limit.
To calculate the limit, often times we can just plug in the limit value into the expression. However, in this case if we were to do that we get , which is undefined.
What we can do to fix this is use L'Hopital's rule, which says
.
So, L'Hopital's rule allows us to take the derivative of both the top and the bottom and still obtain the same limit.
.
Plug in to get an answer of .
Example Question #2 : L'hospital's Rule
Calculate the following limit.
If we plugged in the integration limit to the expression in the problem we would get , which is undefined. Here we use L'Hopital's rule, which is shown below.
This gives us,
.
However, even with this simplified limit, we still get . So what do we do? We do L'Hopital's again!
.
Now if we plug in infitinity, we get 0.
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