11A - An introduction to limits

An introduction to limits

In mathematics, you will come across things that you cannot work out directly. For example, how does the following function behave at x = 1?

Here we run into trouble, we do not know the value of 0/0 (it is "indeterminate"). Therefore, instead of evaluating the function at x = 1 directly, we will approach the value by getting closer and closer to x = 1:

We never reach a limit...

While we may approach 2 from the left hand side and the right hand side of the function, f, we never reach the point we only approach it. Therefore, we may want to state f(1) = 2; however, we cannot because that value doe not exist. Instead, we can say that the limit of f(x) as x approaches 1 is 2. Mathematically, we express this as:

We will explore limits further when we examine continuity and smoothness.

Limits of continuous functions

For any continuous function, f, the limit as x approaches a is equal to f(a); that is:

Continuous function

We will describe continuity in greater detail when we examine continuity and smoothness. However, all functions we have considered so far are continuous over their maximal domain:

• Polynomials (linear, quadratic, cubic and quartic)
• Square root
• Hyperbola (excluding the asymptotes)
• Truncus (excluding the asymptotes)
• Exponential functions
• Logarithmic functions
• Circular functions (sine, cosine and tangent)

Where a function is undefined, the limit will not exist.

10A - VIDEO EXAMPLE 1:

Determine each of the following limits:

Limit theorems

We can use the following theorems when calculating limits.

Theorem 1: Limits of a constant term

The limit of a constant term is simply the value of the constant:

Theorem 2: The sum of limits

Limits that have been added or subtracted can be evaluated separately as their individual limits:

Theorem 3: The product of limits

Limits that have been multiplied can be evaluated separately as their individual limits:

Theorem 4: The quotient of limits

Limits that have been divided can be evaluated separately as their individual limits:

10A - VIDEO EXAMPLE 2:

Determine each of the following limits:

Applying limits to calculus

We introduced the concept of a limit so that we could apply it in our study of calculus. In section 10 we examined average rates of change, where the average rate of change for a function over the interval [a, b] is:

We also examined the instantaneous rate of change by drawing a tangent to a point on a curve and determining the gradient of the tangent. With the knowledge of limits, we can define a tangent as being a secant (line) joining the point x and x + h and as we let h approach 0 the secant becomes a tangent, thus:

...