Question

How do you use the limit definition to find the slope of the tangent line to the graph `f(x)=37` at (0,37)?

Answer 1

I am not sure I got the question right...

Explanation:

Your function is a constant and represents a perfectly horizontal line passing through `y=37`
graph{0x+37 [-84, 82.7, -15.3, 68]}
You can see that this line will have zero slope and I am not sure you can have a tangent to it because you'll need a coincident line-tangent (and a tangent should have only one point in common with your curve)!

You can still find the slope (the derivative) of your original function considering that a limit of a constant is equal to the constant:
Slope`=f'(x)=lim_(h->0)((f(x+h)-f(x))/h)`
but in your case:
`f(x+h)=37`
and
`f(x)=37` as well, being a constant!
So:
Slope`=f'(x)=lim_(h->0)((37-37)/h)=lim_(h->0)((0)/h)=lim_(h->0)(0)=0`