# AP Calculus BC : Euler's Method and L'Hopital's Rule

## Example Questions

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### Example Question #1 : Euler's Method And L'hopital's Rule

Evaluate:

The limit does not exist.

Explanation:

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 : Euler's Method And L'hopital's Rule

Evaluate:

The limit does not exist.

Explanation:

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 #1 : Euler's Method And L'hopital's Rule

Evaluate:

The limit does not exist.

Explanation:

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 #1 : L'hospital's Rule

Evaluate:

Explanation:

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 : Euler's Method

Suppose we have the following differential equation with the initial condition:

Use Euler's method to approximate , using a step size of .

Explanation:

We start at x = 0 and move to x=2, with a step size of 1. Essentially, we approximate the next step by using the formula:

.

So applying Euler's method, we evaluate using derivative:

And two step sizes, at x = 1 and x = 2.

And thus the evaluation of p at x = 2, using Euler's method, gives us p(2) = 2.

### Example Question #2 : Euler's Method

Approximate  by using Euler's method on the differential equation

with initial condition  (which has the solution ) and time step

Explanation:

Using Euler's method with  means that we use two iterations to get the approximation. The general iterative formula is

where each  is

is an approximation of , and , for this differential equation. So we have

So our approximation of  is

### Example Question #1 : Euler's Method And L'hopital's Rule

Evaluate the limit using L'Hopital's Rule.

Undefined

Explanation:

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 #3 : L'hospital's Rule

Evaluate the limit using L'Hopital's Rule.

Undefined

Explanation:

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 #1 : Limits

Evaluate the limit using L'Hopital's Rule.

Undefined

Explanation:

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 #5 : L'hospital's Rule

Calculate the following limit.

Explanation:

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 .

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