When Slopes Become Unbounded
A tangent line usually has a finite slope. But a curve can become increasingly steep near a point, so that the slopes of nearby secant lines grow without bound. In that situation, the curve may have a vertical tangent.
The cube-root function provides a basic example. At \(x=0\), its difference quotient is positive on both sides of zero and grows without bound as \(x\) approaches zero. The graph has a vertical tangent there, even though the function does not have a finite derivative at that point.
The distinction matters: a vertical tangent describes the limiting direction of nearby secant lines, while an ordinary derivative is a finite real number defined by a limit.
Finite Derivatives and Infinite Limits
For a function \(f\) defined near \(a\), the derivative at \(a\), when it exists, is the finite real limit
Here \(h\ne 0\) in the quotient, and the limit must exist as a real number. The expression being limited is the slope of the secant line through \((a,f(a))\) and \((a+h,f(a+h))\).
A quotient can instead tend to \(+\infty\) or \(-\infty\). These describe unbounded behavior, not real-number limits. In particular, if the difference quotient tends to the same infinity from both sides, the nearby secant slopes indicate a vertical tangent.
For a limit of \(+\infty\), the precise meaning is that for every \(M>0\), the quotient is greater than \(M\) whenever \(h\ne 0\) is sufficiently close to zero. For a limit of \(-\infty\), it is less than \(-M\) whenever \(h\ne 0\) is sufficiently close to zero. The same condition must hold on both sides for the two-sided limit to have that value.
Worked Example: The Cube-Root Function at Zero
Worked Example: The Cube-Root Function at Zero
Let \(f(x)=\sqrt[3]{x}\). Determine whether \(f\) has a finite derivative or a vertical tangent at \(x=0\).
First, \(f(0)=0\). For \(h\ne 0\), the difference quotient is
Since \(h=(\sqrt[3]{h})^3\), and \(\sqrt[3]{h}\ne 0\), this simplifies to
This quotient is positive for every \(h\ne 0\). As \(h\) approaches zero from either side, \(|h|^{2/3}\) approaches zero through positive values, so the quotient grows without bound:
For a direct check of the unbounded limit, let \(M>0\) and choose \(\delta=M^{-3/2}\). If \(0<|h|<\delta\), then
Thus the quotient tends to \(+\infty\) from both sides. The function has a vertical tangent line \(x=0\), but it has no finite derivative at zero. As a check, the quotient itself is exactly the secant slope used in the test, and that slope is positive and unbounded for both positive and negative \(h\).
Why an Infinite Slope Gives a Vertical Tangent
The geometric connection comes from the slope of a secant line. If its slope is \(m\), its angle \(\theta\) to the positive horizontal direction satisfies \(\tan(\theta)=m\). As \(m\) grows without bound positively, the line’s direction approaches vertical. As \(m\) decreases without bound negatively, the direction also approaches vertical, with the opposite orientation along that line.
Therefore, when the difference quotient approaches \(+\infty\) or \(-\infty\) from both sides, secant lines through the fixed point and nearby points approach the vertical line through that point. Since the fixed point is \((a,f(a))\), this line has equation \(x=a\).
Use \(\frac{f(a+h)-f(a)}{h}\), with \(h\ne 0\).
Determine whether the quotient approaches the same infinity as \(h\to 0^-\) and \(h\to 0^+\).
If it tends to \(+\infty\) or \(-\infty\) from both sides, report a vertical tangent \(x=a\) and no finite derivative at \(a\).
The geometric explanation supports the classification, but the limit test is decisive: the quotient must have the required two-sided unbounded behavior. A steep-looking graph alone is not enough to establish it.
Worked Example: A Shifted and Scaled Cube Root
Worked Example: A Shifted and Scaled Cube Root
Let \(g(x)=3-2\sqrt[3]{x-1}\). Test for a vertical tangent at \(x=1\), and identify the tangent line if one exists.
At \(x=1\), \(g(1)=3\). For \(h\ne 0\),
Because \(|h|^{2/3}>0\), the quotient is negative on both sides of zero. Its magnitude grows without bound: for any \(M>0\), choose \(\delta=(2/M)^{3/2}\). If \(0<|h|<\delta\), then
Therefore the difference quotient tends to \(-\infty\), and the graph has a vertical tangent at \((1,3)\). The tangent line is \(x=1\). Substituting the point confirms that \((1,3)\) lies on this line. The derivative is not a finite real number because the quotient tends to \(-\infty\), not to a real limit.
Worked Example: An Unbounded Part Plus a Finite Part
Worked Example: An Unbounded Part Plus a Finite Part
Let \(q(x)=\sqrt[3]{x}+x\). Determine the behavior of the difference quotient at \(x=0\).
Since \(q(0)=0\), for \(h\ne 0\),
The first term tends to \(+\infty\) from both sides, while the second term is constant. More explicitly, for any \(M>0\), choose \(\delta=(M+1)^{-3/2}\). If \(0<|h|<\delta\), then
Thus the quotient tends to \(+\infty\). The graph has a vertical tangent at \((0,0)\), with tangent line \(x=0\), and \(q\) has no finite derivative at zero. This example shows directly that adding a bounded contribution to the quotient does not prevent its positive unbounded growth.
Contrast: A Finite Derivative
A large slope near a point is not automatically a vertical tangent. The quotient must actually become unbounded. For comparison, let \(r(x)=x^2\) at \(x=0\). Then \(r(0)=0\), and for \(h\ne 0\),
As \(h\to 0\), this quotient tends to \(0\), a finite real number. Therefore \(r'(0)=0\); the difference quotient does not indicate a vertical tangent. Substituting \(h=0\) into the simplified quotient gives \(0\), confirming the limiting value, not the original quotient’s value at \(h=0\), where it is undefined.
Common Mistake: Calling Infinity the Derivative
A tempting but incorrect conclusion for the cube-root function is:
The calculation of the limit is correct, but the conclusion misuses the definition. A derivative, when it exists, is a finite real number. The correct statement is that the difference quotient tends to \(+\infty\), so the finite derivative \(f'(0)\) does not exist, and the graph has a vertical tangent \(x=0\).
Another important check is that the behavior must agree from both sides. A quotient that tends to \(+\infty\) on one side and \(-\infty\) on the other does not tend to one infinity as a two-sided limit. That situation is not classified by the same two-sided vertical-tangent test.
Why the Distinction Matters
The derivative records an instantaneous rate of change as a finite number. A vertical tangent instead records a limiting direction of the graph when nearby secant slopes become unbounded. Keeping the two statements separate lets you describe the graph accurately without assigning a value that is outside the definition of a derivative.
When analyzing a candidate point, first evaluate the difference quotient, then check the signs and limits from both sides. If both sides approach the same infinity, report the vertical tangent line and state that there is no finite derivative at the point.
Check Your Understanding
For each question, use the difference quotient and distinguish a finite derivative from an unbounded limit.
- For \(f(x)=\sqrt[3]{x}\), simplify the difference quotient at zero and state its two-sided limit.
- If a difference quotient tends to \(-\infty\) from both sides at \(x=a\), what is the vertical tangent line, and does the finite derivative exist?
- For \(g(x)=2+\sqrt[3]{x-3}\), compute the difference quotient at \(x=3\) and classify its limit.
- For \(p(x)=x^2\) at zero, compute the difference quotient and explain why it does not indicate a vertical tangent.
- Why is \(f'(a)=+\infty\) not a correct statement under the definition of a finite derivative?
