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The left-hand derivative of a function at a specific point captures the instantaneous rate of change of the function as the input approaches the given point from the left. In mathematical terms, if we're interested in the left-hand derivative of a function at a point, say, at x=1, we would look at the limit of the difference quotient as h approaches zero from the negative side:
\[ f'_{-}(1) = \blim_{{h \to 0^{-}}} \frac{f(1 + h) - f(1)}{h} \]
In simpler terms, think of h as a tiny step to the left of our point of interest x=1. As the step becomes infinitesimally small, we get closer to the actual slope of the function at exactly x=1, approaching from the left. Consider viewing this as zooming in on the graph at x=1 until the curve looks almost like a straight line, and then measuring the slope of that line.

Conversely, the right-hand derivative refers to the instantaneous rate of change of the function as the input approaches the point from the right. The right-hand derivative at a point x=1 is mathematically represented by the limit of the difference quotient as h approaches zero from the positive side:
\[ f'_{+}(1) = \blim_{{h \to 0^{+}}} \frac{f(1 + h) - f(1)}{h} \]
Again, if h is a minuscule step, this time to the right of our point of interest, as it shrinks to zero, we approach the exact slope at x=1, but from the right. A helpful visualization is to imagine a tiny ant walking towards x=1 from the right side, and just as it's about to reach x=1, we calculate the slope of the path it has traveled. In the exercise provided, neither the left-hand nor the right-hand limit exists due to the square root of a negative number that arises, highlighting the importance of these limits in evaluating differentiability.

The limit of a function encapsulates the value that a function approaches as its input approaches some value. It is crucial in calculus because it helps define both the continuity and differentiability of functions. When we say \( \blim_{{x \to c}} f(x) = L \), we mean that as x gets very close to c, the function f(x) gets arbitrarily close to L. However, the function doesn't necessarily have to reach L when x=c.

In terms of differentiability, if we cannot define a limit as we approach from either the left or the right side of a point, as with the square root function in our exercise, the concept indicates that the function may have a corner, cusp, or discontinuity at that point. In these cases, the function is not differentiable there, as there's no singular, well-defined slope.

Continuity and differentiability are two foundational concepts in calculus that are intimately connected. A function is continuous at a point if you can draw through that point without lifting your pencil. Formally, this means a function f(x) is continuous at a point x=c if \( \blim_{{x \to c}} f(x) = f(c) \). Differentiability, on the other hand, implies that a function has a well-defined tangent or rate of change at that point, which requires the function to be smooth or have no sharp corners or edges.

For a function to be differentiable at a point, it must be continuous there, but the converse isn't necessarily true — a continuous function might still have sharp turns that prevent differentiability. The exercise shows that the lack of left-hand and right-hand derivatives at x=1 results in the function f(x) = \sqrt{1 - x^{2}} being non-differentiable at that point, illustrating that continuity alone is not enough to guarantee differentiability.