1. Gradient of a line passing through two points
2. Limits
3. Differentiation from first principles
4. Differentiating expressions of the form \(kx^n\) with respect to \(x\)
5. The gradient at a point on a curve
6. Tangents and normals

Limits

Roughly speaking, a limit of a function is the value that function approaches as the input into the function approaches some value.

Consider the function \(\text{f}(x)=\dfrac{1}{x}\). As \(x\) gets larger, the value of \(\text{f}(x)\) decreases, getting closer and closer to 0, but never reaching or dropping below 0. We write that \(\displaystyle \lim_{x \to \infty} f(x) = 0\), and say this as follows: “the limit as \(x\) tends to infinity of \(\text{f}(x)\) is 0.”

This applet has four examples. In each case, the aim is to understand what happens to the value of \(\text{f}(x)\) as \(x\) increases. Use the yellow “+1” button to repeatedly increase the value of \(x\) by 1. Once you’ve done this a few times, consider the values of \(\text{f}(100)\text{, f}(1000)\text{, f}(1,000,000)\) etc. to get a sense of the limits reached by these functions.

Compare the above examples with the function \(\text{f}(x)=x^2\). In this case, you can hopefully see that as \(x\) increases, so does the value of \(\text{f}(x)\). There is no limit in this case.