\(\mathrm{f}(\mathrm{x})=\mid \begin{array}{ll}\left(3 / \mathrm{x}^{2}\right) \sin 2 \mathrm{x}^{2}, & \mathrm{x}<0 \\ {\left[\left(\mathrm{x}^{2}+2 \mathrm{x}+\mathrm{c}\right) /\left(1-3 \mathrm{x}^{2}\right)\right],} & \mathrm{x} \in[0, \infty)-\\{1 / \sqrt{3}\\} \\ 0, & \mathrm{x}=(1 / \sqrt{3})\end{array}\) then in order that \(\mathrm{f}\) to be is continuous at \(\mathrm{x}=0\), value of \(\mathrm{c}\) is: (a) 2 (b) 4 (c) 6 (d) 8

Understanding the concept of limits is essential in calculus and helps us analyze a function's behavior as it approaches a particular point. A limit investigates what value a function approaches as the input (or 'x value') gets closer to some number. Quite often, we are interested in what happens as the function approaches a point where it isn't necessarily defined or where it behaves erratically.
For instance, if we have a function like \( f(x) = \frac{1}{x} \) and we want to know the behavior as x approaches 0, we can say \( \lim_{x \to 0} \frac{1}{x} \) is concerning because the function grows infinitely large. This challenges the notion of finding a precise value, instead leading us to the notion of infinity as the limit. The concept allows for a nuanced discussion of the behavior of functions at points that aren't straightforward to evaluate directly.

For a function to be continuous at a point, several conditions must be satisfied. First, the function must be defined at that point. Second, the limit of the function as it approaches that point from both directions must exist. Lastly, the limit value must equal the function's value at that very point. In other words, you can draw the graph of a continuous function without lifting your pencil from the paper.
Continuity is a significant concept because it guarantees the smooth behavior of functions. It's particularly relevant for solving real-world problems where sudden jumps or breaks in functions would lead to unreasonable or undefined results. One common issue with continuity arises at points where function pieces are stitched together, such as the piecewise function in the exercise. Ensuring that these pieces line up perfectly without jumps or holes is an essential part of making a function continuous at those particular connecting points.

The left-hand limit of a function as x approaches a particular value, say \( a \), is what the function values approach as you come closer to \( a \) from the left side, meaning from the smaller values of x. For example, with \( \lim_{x \to a^{-}} f(x) \), the '-' superscript indicates we're looking at the left-hand limit.

Observing Left-hand Limits

To properly analyze left-hand limits, it's important to consider how the function behaves as x gets infinitesimally close to \( a \) from the smaller side. This can be tricky when the function's structure changes drastically near that point. One should note that the limit itself doesn't have to exist for us to talk about the left-hand limit; it's an exploration of the function's tendency as x approaches from one particular direction only.

In contrast, a right-hand limit looks at the approaching value of a function from the right side, which is to say, from values of x that are bigger than \( a \) when we are considering the limit \( \lim_{x \to a^{+}} f(x) \).

Inspecting Right-hand Limits

Similar to the left-hand limit, when evaluating the right-hand limit, we focus on how the function values get closer to a specific value (could be infinity) as x approaches \( a \) from greater values. This direction-specific approach is crucial when we deal with piecewise functions or functions that aren't symmetric around the point of interest. Understanding both left-hand and right-hand limits is key to discussing continuity at a point and ensuring that a function doesn't have a disconnection or jump.