## Limits and Continuity

The process of finding a limit is a key part of differentiation and calculus. Many limits are difficult to evaluate and we will only be studying some of the more basic limits.

### Limits

Finding the limit of a function at a certain value involves investigating the value of the function as x approaches this value.

This can be done using algebra or by looking at the graph of the function.

For the function f(x) = x + 4

 x f(x) 2.9 6.9 2.99 6.99 3 7 3.01 7.01 3.1 7.1

We say On the graph of f(x) = x + 4 In words, as the value of x approaches 3, the value of the function approaches 7.

To evaluate a limit of this type the x-value is substituted straight into the function.

Often this is not possible and the function needs to be simplified or written in another form first.

 Example Evaluate The x-value of 2 cannot be substituted into the function as the denominator would become 0 and division by 0 is not possible. So the function is factorised and simplified and the x-value of 2 can then be substituted. Some limits have to be found as x gets larger and larger, we refer to this as letting "x tend to infinity".

Example On a graph, this would be the y-asymptote of the hyperbola. As x gets larger the y-value approaches 3.

Note that as x approaches -1, there would be NO LIMIT as the line x = -1 is an asymptote and the y-value on either side of the asymptote gets very large or very small. Algebraically, this is recognised by the expression "non zero over zero" when a direct substitution is attempted.

### Continuity

A function is continuous at all points, if the graph can be drawn without lifting your pen off the page!

In other words there are no breaks in the graph.

e.g. This function is not continuous at x = 4 as f(4) = 7 when A hyperbola would not be continuous where the vertical asymptotes occur. See the graph of the hyperbola above.

### Differentiability

Before we study differentiation, we need to establish when a function can or cannot be differentiated. A function cannot be differentiated:

a. at any holes in the domain.

b. at any point where the gradient is significantly different on either side of the point.

c. where the function is undefined.

 Function a. b. c. Differentiability Function is not differentiableat x = -1 Function is not differentiableat x = -1 or at x = 1 Function is not differentiableat x = -1 Continuity Function is not continuousat x = -1 Function is continuous for all values of x Function is not continuousat x = -1

Note A function can be continuous at a point but not differentiable e.g. Graph b. above.