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Understanding the concept of limits is foundational in calculus and speaks to the very heart of mathematical analysis. A limit captures the behavior of a function as the input gets arbitrarily close to a certain value. But how does this relate to continuity? A function is said to be continuous at a point if the limit of the function at that point equals the function's value at that point. In other words, the graph of a continuous function at that point has no holes, jumps, or breaks.

When we say we're finding the limit as in the exercise, \[ \lim _{x \rightarrow -3}(2 x+5) \], we are asking, 'What value does the function approach as the variable 'x' approaches -3?' For linear functions—straight lines like this exercise—the limit as x approaches any value is simply the value of the function at that point, which indicates that linear functions are always continuous at every point unless they have been artificially defined to have a discontinuity like a hole.

Evaluating limits is a skill that allows you to determine the value that a function approaches as the input approaches a certain number. Depending on the complexity of the function, there are various methods to determine limits, such as direct substitution, factoring, rationalizing, and using special limit laws.

In the case of our example, since we are dealing with a linear function, we can evaluate the limit using direct substitution. Direct substitution means plugging the target x-value, in this case, -3, directly into the function. So, by replacing 'x' in the equation \(2x+5\) with -3, we are left with \(2(-3) + 5\), which simplifies to -1. It's the simplest method but also the most powerful in cases where the function is well-behaved, as it is with most linear functions.

Linear functions are the simplest of functions with a straight-line graph. They have the general form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. Linear functions are always continuous and differentiable except at the points of discontinuity introduced by definition.

The exercise provided involved a linear function, specifically the equation \(2x + 5\). When graphed, it's a straight line, which tells us that, generally, the function lacks any sort of complexity that would inhibit limit evaluation. This simplicity is quite powerful as it assures us that evaluating the limit of a linear function at any given point is as simple as plugging the value of 'x' into the equation. There are no limits to the left or the right that differ, as is the case with more complex functions—just a single, unambiguous answer.