Regarding the calculation of the slope of a tangent.

Question

I have recently had some of my first lessons in calculus. We've learned to use the well-known formula for the slope of a tangent:

$$m_t=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$

When working with this formula, I was told that the equation on the left-side should be simplified such that when $h=0$, there would be no division by $0$.

Obviously, I'm extremely new to these calculus concepts.

I want to know what happens when the equation cannot be simplified and there is a division by $0$. That is, there is no solution. What does a situation like this represent? What would it look like graphically?

Answer 1

There is no division by zero, because you're not evaluating the $\dfrac{f(a+h)-f(a)}{h}$ at zero. Instead, you're taking a limit, i.e. you're trying to figure out what this expression approaches, as $h$ approaches zero.

There are some times when this limit is not defined. For example, let $f(x)=\sqrt[3]{x}$ and let $a=0$. Then

$$\dfrac{f(a+h)-f(a)}{h}=\dfrac{\sqrt[3]{a+h}-\sqrt[3]{a}}{h}=\dfrac{\sqrt[3]{h}}{h}=\dfrac{1}{\sqrt[3]{h^2}}.$$

And this expression will approach $\infty$ as $h$ approaches $0$.

Answer 2

There is no graphic distinction between those cases where you can simplify away the $h$, and those cases when you can't. The distinction is a purely algebraic one.

For instance, if $f(x) = x^3$, then $$ \frac{f(x+h) - f(x)}{h} = \frac{(x+h)^3 - x^3}{h} =\\ \frac{3x^2h + 3xh^2+h^3}{h} = 3x^2 + 3xh + h^2 $$ (as long as $h\neq 0$). So we can simplify away the $h$.

On the other hand, for $g(x) = e^x$, we have $$ \frac{g(x+h) - g(x)}{h} = \frac{e^{x+h} - e^x}{h}\\ = e^x\cdot \frac{e^h - 1}{h} $$ and this can't go any further. You can't get rid of the $h$ in the denominator. You just have to know that $e$ is defined specifically to be the number that makes the limit of the final fraction (as $h\to 0$) equal to $1$. And while yes, $x^3$ and $e^x$ are visually distinct, there is nothing about their visual differences that corresponds directly to the fact that one can be simplified and the other not.