Can a tangent line cross the curve?

Yes. At an inflection point, the tangent line crosses the curve. For example, f(x)=x³ at x=0 has tangent line y=0, which crosses the curve. The definition is about local approximation, not global uniqueness.

What happens when the limit does not exist?

If the two-sided limit of secant slopes does not exist or is infinite, the function is not differentiable at that point. No unique tangent line exists. Examples include corners, cusps, and vertical tangents where the slope goes to ±∞.

Why does the calculator sometimes give a wrong slope?

The TI-84 Plus nDeriv( function uses a symmetric difference quotient with h=0.001. At a non-differentiable point like f(x)=|x| at x=0, it returns 0 instead of undefined. The calculator does not check differentiability; you must verify the limit yourself.

How do I know if my function is differentiable at a point?

Check that the left-hand limit and right-hand limit of (f(a+h)-f(a))/h exist and are equal. If they match, the derivative exists. If they differ or go to infinity, the function is not differentiable at that point.

Why a Tangent Slope Is a Limit with a Worked Example

Q: What limiting process turns nearby secant lines into one instantaneous tangent?

The limit as h→0 of the secant slope (f(a+h)-f(a))/h. As h approaches zero, the second point slides toward the first, and the secant line pivots into the tangent line. The limit of these secant slopes is the instantaneous slope of the tangent.

The honest definition of a tangent slope is a limit, and here is why a tangent slope is a limit: the slope of the tangent line at a point is the limit of the slopes of secant lines as the second point approaches the first. No limit, no tangent. The secant slope (f(a+h)-f(a))/h is undefined at h=0, but the limit as h→0 gives the instantaneous slope you need. Work one example and you will see the limiting process turn nearby secant lines into one instantaneous tangent.

Why a Tangent Slope Is a Limit Example: f(x)=x³ at x=1

Take f(x)=x³ at a=1. The point of tangency is (1,1). Pick a second point nearby: (1+h, (1+h)³). The secant line through these two points has slope ((1+h)³ - 1)/h. Expand (1+h)³ = 1 + 3h + 3h² + h³. Subtract 1, divide by h, and you get 3 + 3h + h². This is the secant slope for any h ≠ 0.

Now let h shrink. At h=0.1, the secant slope is 3 + 0.3 + 0.01 = 3.31. At h=0.01, it is 3 + 0.03 + 0.0001 = 3.0301. At h=0.001, it is 3 + 0.003 + 0.000001 = 3.003001. The sequence approaches 3. The limit of secant slopes as h→0 is 3, so the tangent slope is 3. The tangent line at (1,1) is y-1=3(x-1) or y=3x-2.

This is the limiting process: as h→0, the second point (1+h, f(1+h)) slides along the curve until it merges with (1,1), and the secant line pivots into the tangent line. Without the limit, you have no line through one point; with the limit, you have the instantaneous line.

Calculation Steps

The limit definition of the derivative is f'(a) = lim_{h→0} (f(a+h)-f(a))/h. This is not a slogan; it is the calculation you run. For f(x)=x³ at a=1, you computed the limit analytically by simplifying the difference quotient and then letting h go to zero. The simplification step is crucial because you cannot divide by zero. The limit supplies the definition rather than an unrelated calculus mention: the derivative exists exactly when this limit exists and is the same from both sides.

If you need to check your work at this point, a limit calculator can confirm the numeric approach. But the definition itself is the limit, not the calculator's approximation. The slope 3 comes from the limit, not from a machine.

Limits and Failure Cases

The limiting process turns nearby secant lines into one instantaneous tangent. A secant line connects two distinct points on the curve. Its slope is the average rate of change over the interval [a, a+h]. As h shrinks, the secant line's slope changes. The limit of these slopes as h→0 is the instantaneous rate of change at exactly x=a.

This process only works when the function is differentiable at that point. Differentiability means the two-sided limit exists and is finite. If the left-hand limit and right-hand limit differ, or if the limit goes to infinity, no unique tangent line exists in the usual sense.

Failure Case: f(x)=x^(2/3) at x=0

Consider f(x)=x^(2/3) at a=0. The secant slope is (h^(2/3) - 0)/h = h^(2/3) / h = h^(-1/3). As h→0 from the right, h^(-1/3) → +∞. As h→0 from the left, h^(-1/3) → -∞ (since h is negative, raising to a fractional power gives a negative real number for odd denominators). The two-sided limit does not exist as a finite number. The secant slopes blow up to infinity, which corresponds to a vertical tangent at x=0. The tangent line is x=0, slope undefined.

This failure case shows the limit must exist and be the same from both sides. If it does not, the function is not differentiable at that point, and no single tangent slope exists. The limiting process fails because the secant lines do not converge to a unique line.

Common Misconception: The Tangent Line Touches Exactly Once

Many newcomers believe a tangent line touches the curve at exactly one point and never crosses it. This is false. For f(x)=x³ at x=0, the tangent line is y=0. That line crosses the curve at x=0 and intersects again at other points. At an inflection point, the tangent line crosses the curve. The honest definition is the limit of secant lines, not a statement about global intersection count.

Who This Suits and Who Should Skip It

This explanation suits AP Calculus AB/BC students who need to compute tangent lines by hand for the exam, college Calculus 1-2 students who need tangent lines for curve sketching and linear approximations, calculus tutors who need to explain why the touches-without-crossing definition fails, and self-studying calculus learners who need a reliable method. Anyone who does not know what a derivative is or how to compute basic limits should first study the limit definition of the derivative and the power rule from OpenStax Calculus Vol. 1 sections 2.1, 3.1, and 3.3. You can differentiate polynomial, trigonometric, exponential, and logarithmic functions. The single thing that most often goes wrong is assuming the calculator's nDeriv( function gives the correct slope at non-differentiable points; it does not check differentiability, and at a corner or cusp it can return a false slope.