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Understanding limits is the cornerstone of calculus. In essence, a limit is a way to describe the behavior of a function as the input (or variable) gets extremely close to a certain value. This concept is fundamental because it allows us to handle situations that are otherwise challenging, like dealing with values that approach infinity or those that are not clearly defined at a specific point.

When we write the expression \( \text{{lim}}_{x \to c} f(x) = L \), what we're saying is that as x gets infinitely close to the value c, the function \( f(x) \) gets infinitely close to the value L. This doesn't necessarily mean that \( f(x) \) will equal L when x equals c – it is all about the behavior as we approach that value. The limit can be finite, infinite, or it might not exist at all if the function behaves erratically as it approaches c.

The process of finding the value a function approaches as the input approaches a certain value is called approaching a limit. During this approach, we're not necessarily interested in the function's exact value at that point, but rather the value it gets closer to as the input gets ever so close to the target.

Visualizing this on a graph can be helpful: as you draw a line following the graph of the function, your pencil traces a path that homes in on a certain y-value as x nears a certain number. The process is what defines continuity in functions—if there's no drastic change, break or bounce at that point, then you can say that the function is continuous, which is an important concept in calculus and essential in defining the function's limits.

Within the realm of limits, unilateral limits are those where the approach to a specific value is from one side only—either from the left (left-hand limit) or from the right (right-hand limit). These are crucial when assessing the behavior of functions that aren't 'nice and smooth' at a point. In the original exercise, the left-hand limit, denoted by \( \lim_{x \to c^{-}} f(x) = L_1 \), is under examination. This represents the limit of \( f(x) \) as x approaches c exclusively from values less than c.

If you were to plot the function and follow its graph from the left towards the point c, wherever the y-value settles as you get arbitrarily close to c, that's your left-hand limit, \( L_1 \). It’s significant to evaluate unilateral limits since they reveal whether a function has a jump discontinuity or behaves differently when approached from opposite directions. Often, a function's limit at a point exists only if the left-hand and right-hand limits are equal; if they aren't, we say the overall limit does not exist.