# Limits and Imp/Explicit Differentiation

1. Oct 26, 2005

### Sesse

*opps fixing the latex code*
I have a test coming up so I tried solving some review questions but couldn't solve all of them. I will also post the ones which I am not sure whether they are correct or wrong so please spare some time to check them too.
1.
$$F(x)= \left{ \begin{array}{cc} -x, & \mbox{ if } x<-1\\ x^2, & \mbox{ if } -1 \leq x<1\\ \frac{1}{x}, & \mbox{ if } x\geq 1 \end{array} \right$$
Use limit definition of the derivative to see if the fuction is differentiable at $$x=-1$$ and $$x=1$$
At $$x=-1$$
$$\lim_{x\rightarrow -1^-} \frac{f(x)-f(-1)}{x-(-1)}=-1$$
$$\lim_{x\rightarrow -1^+} \frac{f(x)+1}{x+1} = \frac{(x+1)(x-1)}{x+1}=x-1=-2$$
If this solution for $$x=-1$$ is correct then it is also correct for $$x=1$$ so I won't show my solution for that.
So the fuction is not differentiable at both points?
2. Probably correct.
3. $$y=(t^{2/3}).e^{sin(t)}+t$$
Find the slope at (0,0)
$$\frac{dy}{dx}=\left(\frac{2}{3}t^{-1/3}e^{sin(t)}+t^{2/3}e^{sin(t)}cos(t)\right)+1$$
$$\frac{dy}{dx}=1?$$

More will come. :)

Last edited: Oct 26, 2005
2. Oct 26, 2005

### HallsofIvy

Staff Emeritus
Yes, both of those are correct.