# Existense of Tangent Line

1. Jun 26, 2014

### PsychonautQQ

1. The problem statement, all variables and given/known data
Q: Does the graph of f(x) = x^(1/2) have a point of tangency with the line y = (x/4) + 1?

2. Relevant equations
lim x->a (f(x) - f(a)) / (x-a)

3. The attempt at a solution
If the limit exists of the relevant equation than there is a point of tangency.

So i'm having a bit of trouble proving anything here....
using the equation I come to

lim x->a (x^(1/2) - (x/4) + 1) / (x-a)

which looks like as x goes to a the denominator will approach zero which leads me to believe it diverges to infinity??? does that mean the limit doesn't exist?

When setting the equations equal to eachother I find that they intersect at (4,2). Do I plug 4 in for a? same thing happens except this time the numerator obviously approaches zero as well...

2. Jun 26, 2014

### CAF123

What does the curve y=√x and a line of tangency have in common? You do not have to resort to the precise definition of a limit to solve the question.

3. Jun 26, 2014

### PsychonautQQ

How would I use the precise definition here? Just for the sake of using it. To answer your question the curve y=x^(1/2) and a tangent line have in common? I'm not sure... This equation will cover the whole positive y positive x axis as x-> infinity but will get more and more linear looking at big values of x? And tangent lines are linear?

4. Jun 26, 2014

### ehild

5. Jun 26, 2014

### CAF123

A tangent line to a curve at a point has the same gradient as the curve locally.
Try writing the above in math. You are right that the line and the curve intersect only at x=4, so that would be the only possible point of tangency. You want the limit as x → 4 of f(x) = √x to be the same as the limit as x→4 of g(x) = (x/4) + 1.

See above.

You could do the same as the above, but without invoking the limit. What is the derivative (i.e gradient) of y=√x at any point along the real axis? What about y=(x/4)+1?

6. Jun 27, 2014

### CAF123

This should read: you want the limit as x→4 of (f(x)-f(4))/(x-4) and the limit as x→4 of (g(x)-g(4))/(x-4), where f and g are defined in the quote.