Limits

• Limits form the basis of calculus and are a central theme in the Further Pure Mathematics 1 syllabus.
• The limit of a function is the value that the function approaches as the input (or variable) approaches a certain value.
• To solve a limit, one method is to simply substitute the value that the variable is approaching into the function. However, this direct substitution doesn’t always work, especially when complex numbers, infinity, or indeterminate forms are involved.
• If a function is continuous at a certain point, its limit at that point is equal to its actual value at that point.
• Limits can be asymptotic. This is when a function approaches a certain value as the variable tends towards infinity or negative infinity. Often, these are represented as a dashed line, or an asymptote, on a graph.
• Familiarize yourself with the properties of limits. The limit of a constant times a function falls the constant outside the limit. Also, the limit of a sum is equal to the sum of the limits. Understanding these rules can simplify and quicken your calculations.
• If a limit results in an indeterminate form, such as 0/0, you may apply L’Hospital’s Rule where you take the derivative of the numerator and the derivative of the denominator and try to solve the new limit.
• To evaluate the limit of a function that consists of a root, you can rationalise the denominator by multiplying the function by a suitable form of 1 to remove the square root from the denominator.
• The squeeze theorem is another useful concept to remember. It states that if a function lies between two other functions (that tend as x tends to certain value towards the same limit), then the limit of the middle function is also equal to this common limit.
• Understanding limits is fundamental to grasping sequences, as it addresses the question of the value that the sequence tends towards.