Limits: Graphically and Numerically

1. \({\displaystyle \lim_{x\to -7^-} f(x)=-2}\)
2. \({\displaystyle \lim_{x\to -7^+} f(x)=2}\)
3. \({\displaystyle \lim_{x\to -4^-} f(x)=2}\)
4. \({\displaystyle \lim_{x\to -4^+} f(x)=2}\)
5. \({\displaystyle \lim_{x\to 0^+} f(x)=0}\)
6. \({\displaystyle \lim_{x\to 2^-} f(x)=2}\)
7. \({\displaystyle \lim_{x\to 2^+} f(x)=2}\)
8. \({\displaystyle \lim_{x\to 5^-} f(x)=5}\)

1. \({\displaystyle \lim_{x\to -7} f(x) }\) DNE
2. \({\displaystyle \lim_{x\to -5} f(x)=2}\)
3. \({\displaystyle \lim_{x\to -4} f(x)=2}\)
4. \({\displaystyle \lim_{x\to -3} f(x)=3}\)
5. \({\displaystyle \lim_{x\to -2} f(x)=2}\)
6. \({\displaystyle \lim_{x\to 0} f(x)=0}\)
7. \({\displaystyle \lim_{x\to 2} f(x)=2}\)
8. \({\displaystyle \lim_{x\to 5} f(x)=5}\)

To find the limit numerically, you need to construct a table similar to the following.

\(x    \rightarrow 1^-\)	\({f(x) =\dfrac{\dfrac{1}{x}-1}{x-1}}\)	\(x    \rightarrow 1^+\)	\({f(x) =\dfrac{\dfrac{1}{x}-1}{x-1}}\)
0.9	-1.1111	1.1	-0.9091
0.99	-1.0101	1.01	-0.9901
0.999	-1.0010	1.001	-0.9990
0.9999	-1.0001	1.0001	-0.9999

From the table, it looks like the \(y\)-values are approaching \(-1\) as \(x\) approaches 1 from the left and from the right, so that's the solution to the limit.

\(\displaystyle {\lim_{x\to 1^-} \dfrac{\dfrac{1}{x}-1}{x-1}} = -1\)