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Limits are a fundamental concept in calculus that help us understand the behavior of functions as they approach a certain point. Imagine limits as if you are zooming in on a function at a specific point to see what value it approaches, even if the function doesn't actually reach that value. For example:

• If you have the function \( f(x) = \frac{\sin x}{x} \), it is undefined at \( x = 0 \) because you cannot divide by zero.
• To find out what happens as \( x \) gets very close to zero, we calculate the limit: \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \). This means that as \( x \) approaches zero, the value of \( f(x) \) gets closer and closer to 1.

Understanding limits is crucial not just for handling points where functions are undefined, but also for calculating derivatives and integrals. Calculating limits often requires using different techniques, such as L'Hôpital's rule, especially when dealing with indeterminate forms like \( \frac{0}{0} \). For instance, when evaluating \( \lim_{x \to 0} \frac{1 - \cos x}{x} \), applying L'Hôpital's rule helps simplify this limit to 0.

Discontinuities occur in functions where the graph has a jump, a break, or a hole. At these points, the function may be undefined or not behave as expected. Typically, discontinuities are classified into different types: removable, jump, and infinite discontinuities. Let’s focus on:

• Removable Discontinuities: These happen when a function is not defined at a point, but if the function could be defined at that point, it removes the break. It's like there's a small hole in the graph that we can fill by redefining the function at that specific point. A common example is \( f(x) = \frac{\sin x}{x} \) where at \( x = 0 \), if we set \( f(0) = 1 \), the function becomes continuous.
• Jump Discontinuities: Occur when there's a sudden change in the value of the function. The function jumps from one value to another.
• Infinite Discontinuities: These discontinuities occur where the function goes to infinity or negative infinity.

Understanding discontinuities helps determine how functions behave and whether they are suitable for further analysis, like integration. It can also inform how to modify or complete the function to ensure it is continuous.

Removable discontinuities are interesting because they occur when a function has a point of discontinuity that can be effectively "removed" by redefining the function at that specific point. Consider it like "filling in" a hole in the graph of the function.

• In mathematical terms, if a function \( f(x) \) has a limit \( L \) as \( x \) approaches \( c \), but \( f(c) \) is not defined or not equal to \( L \), it is a removable discontinuity.
• For example, in the case of \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \), the function can be redefined to make it continuous if we define \( f(0) = 1 \).
• Another example is \( \frac{1 - \cos x}{x} \). At \( x = 0 \), this is undefined, but the limit is 0. By defining the function such that \( g(0) = 0 \), the discontinuity is removed.

Recognizing and addressing removable discontinuities can be especially helpful in calculus, as it allows seamless integration and differentiation of functions that initially appear to be problematic due to undefined points.