12.1 Finding limits: numerical and graphical approaches  (Page 2/10)

Understanding left-hand limits and right-hand limits

We can approach the input of a function from either side of a value—from the left or the right. [link] shows the values of

as described earlier and depicted in [link] .

Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in [link] are $6.9,$ $6.99,$ and $\mathrm{6.999.}$ The corresponding outputs are $7.9,7.99,$ and $\mathrm{7.999.}$ These values are getting closer to 8. The limit of values of $f\left(x\right)$ as $x$ approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function $f(x)=x+1,x\ne 7$ as $x$ approaches 7.

Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in [link] are $7.1,$ $7.01,$ and $\mathrm{7.001.}$ The corresponding outputs are $8.1,$ $8.01,$ and $\mathrm{8.001.}$ These values are getting closer to 8. The limit of values of $f\left(x\right)$ as $x$ approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function $f(x)=x+1,x\ne 7$ as $x$ approaches 7.

[link] shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input $x$ within the interval $6.9<x<7.1$ to produce an output value of $f\left(x\right)$ within the interval $7.9<f(x)<\mathrm{8.1.}$

We also see that we can get output values of $f(x)$ successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.

[link] provides a visual representation of the left- and right-hand limits of the function. From the graph of $f(x),$ we observe the output can get infinitesimally close to $L=8$ as $x$ approaches 7 from the left and as $x$ approaches 7 from the right.

To indicate the left-hand limit, we write

To indicate the right-hand limit, we write