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The limit definition of a derivative is a fundamental concept in calculus. It allows us to determine the instantaneous rate of change of a function at any given point. For any function \( f(x) \), the derivative at a particular point \( x = a \) is given by the limit: \[ f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} \] This formula essentially computes the slope of the tangent line to the curve \( f(x) \) at \( x = a \). In the original exercise, we applied this definition to \( f(x) = \sqrt{x} \) at \( x = 0 \). By substituting, we found: \[ f'(0) = \lim_{h \rightarrow 0} \frac{\sqrt{h}}{h} \] This expression requires us to examine the behavior of \( \frac{\sqrt{h}}{h} \) as \( h \) approaches zero to decide on the existence of the derivative. Using actual numbers for \( h \) allows us to approximate this limit, but as seen with decreasing \( h \), the result keeps increasing, hinting at some underlying math issue.

A derivative might not exist at a certain point like in our function example \( f(x)=\sqrt{x} \) at \( x=0 \). Let’s explore why. There are several reasons a derivative may not exist at a point:

• The function might be discontinuous at that point.
• The function may have a sharp corner or cusp.
• There might be a vertical tangent line with infinite slope.

For \( f(x)=\sqrt{x} \), it's the cusp scenario. This arises because the expression \( \lim_{h \rightarrow 0} \frac{\sqrt{h}}{h} \) does not settle down to a finite value. As we calculated: - For \( h = 0.1 \), the result was approximately 3.162. - For smaller \( h \) values \( 0.001 \) and \( 0.00001 \), the results escalated to 31.622 and 316.228, respectively. This indicates the slope approaches infinity as it stretches to smaller \( h \). Therefore, the derivative—meaning the slope of the tangent—does not exist at that point.

Graphing a function is a powerful way to visualize concepts in calculus, such as derivatives. The function \( f(x) = \sqrt{x} \) presents something particularly important when analyzed visually on a graph. In the graph of \( f(x) = \sqrt{x} \) over the window \([0, 1]\) by \([0, 1]\), you can spot why the derivative at \( x = 0 \) doesn't exist. The graph shows a distinctive cusp at \( x = 0 \). Here’s what to look for:

• At \( x = 0 \), instead of a smooth curve or gentle turning point, you see a sharp corner.
• Such sharp corners or cusps suggest infinite and undefined slopes.
• This graphical behavior aligns perfectly with the algebraic concept we explored, where the slope doesn't stabilize.

Thus, an important lesson is learned: graphical analysis can be used to confirm the characteristics suggested by the calculus of derivatives. Through visuals, we better grasp why at a cusp, a derivative is non-existent.