Requirements for a Tangent at the Origin: Function Analysis

[Jan Hill]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

Homework Statement

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?

Homework Equations

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

 

[Jan Hill]

What is a finite derivative?

 

[LCKurtz]

What is a finite derivative?

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

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

for your function

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