Active Calculus

In Section 1.2, we learned that \(f\) has limit \(L\) as \(x\) approaches \(a\) provided that we can make the value of \(f(x)\) as close to \(L\) as we like by taking \(x\) sufficiently close (but not equal to) \(a\text{.}\) If so, we write \(\lim_{x \to a} f(x) = L\text{.}\)

Essentially there are two behaviors that a function can exhibit near a point where it fails to have a limit. In Figure 1.7.3, at left we see a function \(f\) whose graph shows a jump at \(a = 1\text{.}\) If we let \(x\) approach 1 from the left side, the value of \(f\) approaches 2, but if we let \(x\) approach \(1\) from the right, the value of \(f\) tends to 3. Because the value of \(f\) does not approach a single number as \(x\) gets arbitrarily close to 1 from both sides, we know that \(f\) does not have a limit at \(a = 1\text{.}\)

For such cases, we introduce the notion of left and right (or one-sided) limits.

For the function \(g\) pictured at right in Figure 1.7.3, the function fails to have a limit at \(a = 1\) for a different reason. While the function does not have a jump in its graph at \(a = 1\text{,}\) it is still not the case that \(g\) approaches a single value as \(x\) approaches 1. In particular, due to the infinitely oscillating behavior of \(g\) to the right of \(a = 1\text{,}\) we say that the right-hand limit of \(g\) as \(x \to 1^+\) does not exist, and thus \(\lim_{x \to 1} g(x) \ \text{does not exist} \text{.}\)

To summarize, if either a left- or right-hand limit fails to exist or if the left- and right-hand limits are not equal to each other, the overall limit does not exist.

That is, a function has a limit at \(x = a\) if and only if both the left- and right-hand limits at \(x = a\) exist and have the same value.

In Preview Activity 1.7.1, the function \(f\) given in Figure 1.7.1 fails to have a limit at only two values: at \(a = -2\) (where the left- and right-hand limits are 2 and \(-1\text{,}\) respectively) and at \(x = 2\text{,}\) where \(\lim_{x \to 2^+} f(x)\) does not exist). Note well that even at values such as \(a = -1\) and \(a = 0\) where there are holes in the graph, the limit still exists.

Intuitively, a function is continuous if we can draw its graph without ever lifting our pencil from the page. Alternatively, we might say that the graph of a continuous function has no jumps or holes in it. In Figure 1.7.5 we consider three functions that have a limit at \(a = 1\text{,}\) and use them to make the idea of continuity more precise.

With no hole or jump in the graph of \(h\) at \(a = 1\text{,}\) we say that \(h\) is continuous there. More formally, we make the following definition.

Conditions (a) and (b) are technically contained implicitly in (c), but we state them explicitly to emphasize their individual importance. The definition says that a function is continuous at \(x = a\) provided that its limit as \(x \to a\) exists and equals its function value at \(x = a\text{.}\) If a function is continuous at every point in an interval \([a,b]\text{,}\) we say the function is “continuous on \([a,b]\text{.}\)” If a function is continuous at every point in its domain, we simply say the function is “continuous.” Thus, continuous functions are particularly nice: to evaluate the limit of a continuous function at a point, all we need to do is evaluate the function.

This route of substituting an input value to evaluate a limit works whenever we know that the function being considered is continuous. Besides polynomial functions, all exponential functions and the sine and cosine functions are continuous at every point, as are many other familiar functions and combinations thereof.

We recall that a function \(f\) is said to be differentiable at \(x = a\) if \(f'(a)\) exists. Moreover, for \(f'(a)\) to exist, we know that the function \(y = f(x)\) must have a tangent line at the point \((a,f(a))\text{,}\) since \(f'(a)\) is precisely the slope of this line. In order to even ask if \(f\) has a tangent line at \((a,f(a))\text{,}\) it is necessary that \(f\) be continuous at \(x = a\text{:}\) if \(f\) fails to have a limit at \(x = a\text{,}\) if \(f(a)\) is not defined, or if \(f(a)\) does not equal the value of \(\lim_{x \to a} f(x)\text{,}\) then it doesn’t make sense to talk about a tangent line to the curve at this point.

Indeed, it can be proved formally that if a function \(f\) is differentiable at \(x = a\text{,}\) then it must be continuous at \(x = a\text{.}\) So, if \(f\) is not continuous at \(x = a\text{,}\) then it is automatically the case that \(f\) is not differentiable there. For example, in Figure 1.7.5, both \(f\) and \(g\) fail to be differentiable at \(x = 1\) because neither function is continuous at \(x = 1\text{.}\) But can a function fail to be differentiable at a point where the function is continuous?

In Figure 1.7.8, the function has a sharp corner at a point. For the pictured function \(f\text{,}\) we observe that \(f\) is clearly continuous at \(a = 1\text{,}\) since \(\lim_{x \to 1} f(x) = 1 = f(1)\text{.}\)

But the function \(f\) in Figure 1.7.8 is not differentiable at \(a = 1\) because \(f'(1)\) fails to exist. One way to see this is to observe that \(f'(x) = -1\) for every value of \(x\) that is less than 1, while \(f'(x) = +1\) for every value of \(x\) that is greater than 1. That makes it seem that either \(+1\) or \(-1\) would be equally good candidates for the value of the derivative at \(x = 1\text{.}\) Alternately, we could use the limit definition of the derivative to attempt to compute \(f'(1)\text{,}\) and discover that the derivative does not exist. Finally, we can see visually that the function \(f\) in Figure 1.7.8 does not have a tangent line. When we zoom in on \((1,1)\) on the graph of \(f\text{,}\) no matter how closely we examine the function, it will always look like a “V”, and never like a single line, which tells us there is no possibility for a tangent line there.

If a function does have a tangent line at a given point, when we zoom in on the point of tangency, the function and the tangent line should appear essentially indistinguishable¹. Conversely, if we zoom in on a point and the function looks like a single straight line, then the function should have a tangent line there, and thus be differentiable. Hence, a function that is differentiable at \(x = a\) will, up close, look more and more like its tangent line at \((a,f(a))\text{.}\) Therefore, we say that a function that is differentiable at \(x = a\) is locally linear.

To summarize the preceding discussion of differentiability and continuity, we make several important observations.