Notation

Notation

We can now introduce new notation. For the function $$y=\cfrac{(x+6)(x-2)}{x+6}$$, we can write:

$\lim_{x\to -\text{6}} \cfrac{(x+6)(x-2)}{x+6}=-\text{8}.$

This is read: the limit of $$\cfrac{(x+6)(x-2)}{x+6}$$ as $$x$$ tends to $$-\text{6}$$ (from both the left and the right) is equal to $$-\text{8}$$.

Limits

If $$f(x)=x+1$$, determine:

 $$f(-\text{0.1})$$ $$\phantom{xxxxxx}$$ $$f(-\text{0.05})$$ $$f(-\text{0.04})$$ $$f(-\text{0.03})$$ $$f(-\text{0.02})$$ $$f(-\text{0.01})$$ $$f(\text{0.00})$$ $$f(\text{0.01})$$ $$f(\text{0.02})$$ $$f(\text{0.03})$$ $$f(\text{0.04})$$ $$f(\text{0.05})$$ $$f(\text{0.1})$$

What do you notice about the value of $$f(x)$$ as $$x$$ gets closer and closer to $$\text{0}$$?

Example

Question

Write the following using limit notation: as $$x$$ gets close to $$\text{1}$$, the value of the function $$y=x+2$$ approaches $$\text{3}$$.

This is written as:

$\lim_{x\to 1}(x+2)=3$

This is illustrated in the diagram below:

We can also have the situation where a function tends to a different limit depending on whether $$x$$ approaches from the left or the right.

As $$x\to 0$$ from the left, $$f(x)$$ approaches $$-\text{2}$$. As $$x\to 0$$ from the right, $$f(x)$$ approaches $$\text{2}$$.

The limit for $$x$$ approaching $$\text{0}$$ from the left is:

$\lim_{x\to {0}^{-}}f(x)= -\text{2}$

and for $$x$$ approaching $$\text{0}$$ from the right:

$\lim_{x\to {0}^{+}}f(x)= -\text{2}$

where $$0^{-}$$ means $$x$$ approaches zero from the left and $$0^{+}$$ means $$x$$ approaches zero from the right.

Therefore, since $$f(x)$$ does not approach the same value from both sides, we can conclude that the limit as $$x$$ tends to zero does not exist.

As $$x$$ tends to $$\text{0}$$ from the left, the function approaches $$\text{2}$$ and as $$x$$ tends to $$\text{0}$$ from the right, the function approaches $$\text{2}$$. Since the function approaches the same value from both sides, the limit as $$x$$ tends to $$\text{0}$$ exists and is equal to $$\text{2}$$.

Example

Question

Determine:

1. $$\displaystyle\lim_{x\to 1}10$$
2. $$\displaystyle\lim_{x\to 2}(x + 4)$$

Simplify the expression and cancel all common terms

We cannot simplify further and there are no terms to cancel.

Calculate the limit

1. $$\displaystyle\lim_{x\to 1}10=10$$
2. $$\displaystyle\lim_{x\to 2} (x + 4) = 2 + 4 = 6$$

Example

Question

Determine the following and illustrate the answer graphically:

$\lim_{x\to 10}\cfrac{{x}^{2}-100}{x-10}$

Simplify the expression

Factorise the numerator:

$$\cfrac{{x}^{2}-100}{x-10}=\cfrac{(x+10)(x-10)}{x-10}$$

As $$x \to 10$$, the denominator $$(x – 10) \to 0$$, therefore the expression is not defined for $$x=10$$ since division by zero is not permitted.

Cancel all common terms

$\cfrac{(x+10)(x-10)}{x-10}=x+10$

Calculate the limit

\begin{align*} \lim_{x\to 10}\cfrac{{x}^{2}-100}{x-10} &=\lim_{x\to 10} (x + 10) \\ &= 10 + 10 \\ &= 20 \end{align*}