Limit Definition of Derivative

cjaylee
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Homework Statement


Use the limit definition of derivative to determine the derivative of the following function:

f(x) = { sqrt(x^2+1) if x<=0
0 if x>0

Homework Equations



I'm not sure as to why the function is not continuous at x=0, and so it's not differentiable at that point.

The Attempt at a Solution



The left-hand limit and right-hand limit give me 0. And if I plot a graph, the graph hits 0 as it moves from one function to the other.
 
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The left hand limit is "1", not "0"! I would think that would be obvious. What is f(-0.001)?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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