Limits and Continuity Limits

The curve $y=a^{x}$ approaches the x-axis when x approaches very large negative numbers, but never touches it. That is x $\rightarrow -\infty, a^{x} \rightarrow 0$.

We say that the limit of $a^{x}$ as x approaches $-\infty$ is 0. In symbols, we write: $\lim _{ x\rightarrow -\infty }{ { a }^{ x } } =0$

Examples

1. Find $\lim _{ x\rightarrow 0 }{ \cfrac { { x }^{ 2 }+5x }{ x } }$

Substituting x=0 into the function $\cfrac{0}{0}$, which is undefined. Factorising and cancelling help us find the limit. $\lim _{ x\rightarrow 0 }{ \cfrac { { x }^{ 2 }+5x }{ x } } \\ =\lim _{ x\rightarrow 0 }{ \cfrac { x(x+5) }{ x } } \\ =\lim _{ x\rightarrow 0 }{ (x+5) } \\ =5$

2. Find $\lim _{ x\rightarrow 2 }{ \cfrac { x-2 }{ { x }^{ 2 }-4 } }$

Substituting x=2 into the function gives $\cfrac{0}{0}$, which is undefined. $\lim _{ x\rightarrow 2 }{ \cfrac { x-2 }{ { x }^{ 2 }-4 } } \\ =\lim _{ x\rightarrow 2 }{ \cfrac { x-2 }{ (x+2)(x-2) } } \\ =\lim _{ x\rightarrow 2 }{ \cfrac { 1 }{ x+2 } } \\ =\cfrac { 1 }{ 4 }$

Continuity

Many functions are continuous, which means they have a smooth, unbroken curve (or line). However, several discontinuous functions have gaps in their graphs. If a curve is discontinuous at a certain point, we can use limits to find the value that the curve approaches at that point.

Example

Find $\lim _{ x\rightarrow -2 }{ \cfrac { { x }^{ 2 }+x-2 }{ x+2 } }$ and hence sketch the curve $y=\cfrac { { x }^{ 2 }+x-2 }{ x+2 }$.

Substituting $x=-2$ into $\cfrac { { x }^{ 2 }+x-2 }{ x+2 }$ gives $\cfrac{0}{0}$. $\lim _{ x\rightarrow -2 }{ \cfrac { { x }^{ 2 }+x-2 }{ x+2 } } \\ \\ =\lim _{ x\rightarrow -2 }{ \cfrac { (x-1)(x+2) }{ x+2 } } \\ \\ =\lim _{ x\rightarrow -2 }{ (x-1) } \\ \\ =-3$ $y=\cfrac { { x }^{ 2 }+x-2 }{ x+2 } \\ \\ y=\cfrac { (x+2)(x-1) }{ x+2 } \\ \\ y=x-1$
So the function $y=x-1$ where $x\neq -2$. It is discontinuous at (-2, -3).