2.1: Definition of Limit
- Page ID
- 178735
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definition of Limit
For an explicit function \(y = f(x)\), when you plug in \(x = a\), you get one specific output \(f(a)\).
Sometimes, we are curious how a function acts nearby to a instead of directly at a.
This brings us to a very powerful tool of calculus: limits.
We write \(\textcolor{red}{ \displaystyle{ \lim_{x\to a^+} f(x)=L} }\) if: when \(x>a\) is close to \(a\), we must have \(f(x)\) close to \(L\).
Notice that in the picture above, we have \(f(1)=2\), but nothing nearby to \(x=1\) (like \(x=1.1, 1.02, 1.0000783,\) etc) has an output that is close to \(2\).
Instead, when \(x\) is strictly greater than \(1\) but still very close to \(1\), most of the outputs are very close to some negative number \(L\).
This number \(L\) is our limit, not the value of \(2\).
We write \(\textcolor{red}{\displaystyle{ \lim_{x\to a^-} f(x)=L}}\) if when \(x<a\) is close to \(a\), we must have \(f(x)\) close to \(L\).
Again in the picture above, we have \(f(1)=-2\), but nothing nearby to \(x=1\) (like \(x=0.9, 0.99, 0.9999783,\) etc) has an output that is close to \(-2\).
Instead, when \(x\) is strictly smaller than \(1\) but still very close to \(1\), most of the outputs are very close to some positive number \(L\).
This number \(L\) is our limit, not the value of \(-2\).
Remember: the value of \(f(a)\) does not influence the limit at \(x=a\) from either side
Since each of \(\displaystyle{ \lim_{x\to a^+} f(x)}\) and \(\displaystyle{ \lim_{x\to a^-} f(x)}\) require \(x>a\) and \(x<a\) respectively:
\(\displaystyle{ \lim_{x\to a^-} f(x)}\) is the left-handed limit, since we are approaching \(x\) from numbers on its left side.
\(\displaystyle{ \lim_{x\to a^+} f(x)}\) is the right-handed limit, since we are approaching \(x\) from numbers on the right side.
Evaluate \(\displaystyle{ \lim_{x\to 2^+} \left(\frac{1}{2}x-1\right)}\) and \(\displaystyle{ \lim_{x\to 2^-} \left(\frac{1}{2}x-1\right)}\)
Solution
We can sketch the graph and plug in some neighboring values to \(2\):
In situations like the one above, where the right and left sided limit of a function agree at a given point, we just say \(\displaystyle{ \lim_{x\to 2} \left(\frac{1}{2}x-1\right)=0}\)
Evaluate \(\displaystyle{ \lim_{x\to -1^+} g(x)}\) and \(\displaystyle{ \lim_{x\to -1^-} g(x)}\) for the graph below:
Solution
We can see that for numbers to the left of \(x=-1\), we can see that we approach the point \((-1,-2)\)
Therefore we have \(\displaystyle{ \lim_{x\to -1^-} g(x)=-2}\)
And for numbers to the right of \(x=-1\), the function approaches the point \((-1,1)\)
Therefore we have \(\displaystyle{ \lim_{x\to -1^+} g(x)=1}\)
When the right and left sided limit of a function do NOT agree at a given point
as they just did above, we just say \(\displaystyle{ \lim_{x\to -1} g(x)=}\)DNE (does not exist).
So we can officially say that \(\displaystyle{ \lim_{x\to a^-} f(x)= \lim_{x\to a^+} f(x)=L}\) is the only way for \(\displaystyle{ \lim_{x\to a} f(x)=L}\)
(and when they don't match, the double-sided limit does not exist.)
Lets try a few quick examples before determining some shortcuts in the next section:
Evaluate \(\displaystyle{ \lim_{x\to 4} (7-3x)}\)
Solution
| \(x<4\) | \(f(x)=7-3x\) | \(x>4\) | \(f(x)=7-3x\) | |
| 3.9 | -4.7 | 4.1 | -5.3 | |
| 3.99 | -4.97 | 4.01 | -5.03 | |
| 3.99999 | -4.99997 | 4.00001 | -5.00003 |
The left columns show us \(\displaystyle{ \lim_{x\to 4^-} (7-3x)=-5}\) and the right columns give \(\displaystyle{ \lim_{x\to 4^+} (7-3x)=-5}\)
Since these two match, we know that \(\displaystyle{ \lim_{x\to 4} (7-3x)}=-5\)
Evaluate \(\displaystyle{ \lim_{x\to 0} (x^2+2x)}\)
Solution
| \(x<0\) | \(f(x)=x^2+2x\) | \(x>0\) | \(f(x)=x^2+2x\) | |
| -0.1 | -0.19 | 0.1 | 0.21 | |
| -0.01 | -0.0199 | 0.01 | 0.0201 | |
| -0.0001 | -0.000199.. | 0.0001 | 0.00020001 |
The left columns show us \(\displaystyle{ \lim_{x\to 0^-} (x^2+2x)=0}\) and the right columns give \(\displaystyle{ \lim_{x\to 0^+} (x^2+2x)=0}\)
Since these two match, we know that \(\displaystyle{ \lim_{x\to 0} (x^2+2x)}=0\)
Sadly for more complicated functions and numbers, this strategy relies a calculator to be effective.
For the function graphed below, evaluate (a) \(\displaystyle{ \lim_{x\to -2} f(x)}\) and (b) \(\displaystyle{ \lim_{x\to 1} f(x)}\)
Solution
(a) Near \(x=-2\) we see the graph approaches the point \((-2,-2)\) from the left so \(\displaystyle{ \lim_{x\to -2^-} f(x)=-2}\)
and the graph approaches the point \((-2,0)\) from the right side, so \(\displaystyle{ \lim_{x\to -2^+} f(x)=0}\)
Since these two do not match, we know \(\displaystyle{ \lim_{x\to -2} f(x)=}\)DNE
(b) Near \(x=1\) we see the graph approaches the point \((1,3)\) from the left so \(\displaystyle{ \lim_{x\to 1^-} f(x)=3}\)
and the graph approaches the point \((1,3)\) from the right side, so \(\displaystyle{ \lim_{x\to 1^+} f(x)=3}\)
Since these two match, we can say that \(\displaystyle{ \lim_{x\to 1} f(x)=3}\)
When given a graph, calculators can be avoided, but it is difficult to determine your answer if it isn't an integer or if the graph is not perfectly drawn.