Intuitive Definition of a Limit

e.g. ‘approximating the slope of the tangent line’ or ‘approximating the derivative’

Say you have a parabola, and you choose an arbitrary point on it (let’s use P(1, 4) as an example). If we wanted to find the slope of the tangent line at that exact point, we would need one other point to calculate ∆y / ∆x, which is the slope formula.

Because we don’t have another point on the tangent line, we’ll instead pick a second point Q on the parabola that’s very close to P. When we calculate the slope using P and Q, we get the slope of a secant line connecting the two points.

If we repeat this process and choose points Q that are closer and closer to P, the slope of the secant line gets closer and closer to the slope of the tangent line at P.

As Q gets closer and closer to P, the slope of the secant lines approach a specific value. This value is called the limit, and it represents the slope of the tangent line at P.