A drawing illustrating the first definition given would look something like this:
See as x → a {\displaystyle x\to a} , the slope of the line between ( a , f ( a ) ) {\displaystyle (a,f(a))} and ( x , f ( x ) ) {\displaystyle (x,f(x))} approaches the slope of the tangent line at x = a {\displaystyle x=a} .