The graph shows $f(x) = x(x-2)^2 + 1$.
As $x$ gets closer to $a$, the slope of the secant line between $a$ and $x$ tends toward the slope of the tangent line at $a$.
Mathematically, the slope of the secant line between $a$ and $x$ is
$$\frac{\text{Change in }y}{\text{Change in }x} = \frac{f(x)-f(a)}{x-a}.$$
Hence,
$$\text{Slope of tangent line at }a \text{ is equal to } \lim_{x \rightarrow a} \frac{f(x)-f(a)}{x-a}.$$