Recall formula of the tangent line to a two-dimensional function from Calculus 1:

For the Euler's method, this equation can be rewritten as:

Where  is the next value of the solution function,  is the derivative of the previous step. 

We are given a differential equation . Therefore, if we know the value of  and , we can find the . Consecutively, we can plug it into the Euler's method equation and find the .

Here is how this process is executed for the y(1):

1) Identify the initial condition and step size. This is the only information that Euler's method uses to calculate the approximation of solution function, other than the equation itself. In this case, we can say, that for the first step,

Note, that for the whole process, step size remains the same, 

2) Calculate the derivative of function at point  using given differential equation. In this case,

3) Note that in step 2 we found . Now we have all components to plug in to the Euler's method formula. Again, for our first step:

The same algorithm is used for y(2) and y(3). Note, that if we know step size and range of x, we will always know . For each consecutive step,  is the value of the solution function, found in the previous step.

Let's examine the limit

first.

and 

,

so by L'Hospital's Rule, 

Since ,

Now, for each , ; therefore, 

By the Squeeze Theorem, 

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 

But  for any , so 

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

and 

Therefore, by L'Hospital's Rule, we can find  by taking the derivatives of the expressions in both the numerator and the denominator:

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.

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.

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

.

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

. 

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 .