When we talk about the limit definition of a derivative, it’s important to know that sometimes this definition doesn’t give us the right answers or information we need. The derivative is usually written as

$$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h},$$

and it helps us understand how functions behave. But there are special situations where this limit definition doesn’t really work or can even lead us to the wrong conclusion.

First, let’s look at discontinuities. This happens when a function isn’t continuous at a point $a$. If there’s a jump, an infinite break, or the function is broken up in some way, the limit for the derivative might not exist. For example, if you draw the graph of such a function, the left-hand limit and the right-hand limit could give different answers, or one limit might not exist at all. So, if the function isn’t continuous, we can’t use the derivative limit definition properly.

Next, let’s talk about sharp corners or cusps. A well-known case is the absolute value function $f(x) = |x|$ at $x = 0$. If we try to find the derivative

$$f'(0) = \lim_{h \to 0} \frac{|h| - |0|}{h},$$

it doesn’t work because the left-hand limit approaches $-1$, while the right-hand limit approaches $1$. In this case, the limit definition fails because the slope of the tangent line isn’t clear.

Another situation is when there are vertical tangents. Take the function $f(x) = \sqrt[3]{x}$ at $x = 0$. The graph near this point has a vertical tangent. If we try to find the derivative, we might get infinity, which means the slope is undefined. This shows that the usual idea of a derivative doesn’t work well with vertical tangents.

Oscillatory behavior can also make the limit definition of the derivative unreliable. For example, the function $f(x) = \sin(1/x)$ as $x$ gets close to 0 keeps bouncing back and forth between $-1$ and $1$. The limit

$$f'(0) = \lim_{h \to 0} \frac{\sin(1/h)}{h}$$

doesn’t settle on a single number. Because of this, we can’t apply the limit definition here, since it doesn’t give us a clear slope for a tangent line at that point.

Lastly, when we deal with functions that are defined in pieces, there might be points where the limit definition doesn’t work. These functions can suddenly switch from one form to another without being smooth. This change can make it hard to calculate the derivative using the limit definition.

In summary, even though the limit definition of a derivative is a strong tool in calculus, it has some limits that can lead to wrong or unclear outcomes in cases of discontinuities, corners, vertical tangents, oscillatory behavior, and piecewise definitions. It’s really important to carefully look at the function’s behavior before using this formal limit definition for derivatives. Knowing these details helps us understand and use calculus better in real life.