# Tangent at the origin

1. Oct 11, 2010

### Jan Hill

1. The problem statement, all variables and given/known data

What must hold true for a function to have a tangent at the origin.

Eg. Given f(x) = 0, x = 0

and f(x0 = xsin (1/x) x does not equal 0

will the graph have a tangent at the origin?

2. Relevant equations

3. The attempt at a solution

2. Oct 11, 2010

### LCKurtz

It must have a finite derivative when x = 0. Check its difference quotient.

3. Oct 12, 2010

### Jan Hill

What is a finite derivative?

4. Oct 12, 2010

### LCKurtz

Have you had or are you taking calculus? If so then you should know what a derivative is. A function has a finite derivative at a point if its derivative at that point exists and is finite.

Geometrically this means that the graph of the function is smooth enough at the given point that it has a tangent line that is not vertical.

To work this problem you need to analyze

$$\lim_{h\rightarrow 0}\frac{f(0+h)-f(0)} h$$

$$f(x) = x\sin\frac 1 x,\ f(0)=0$$