If we ignore the DNE, we can clearly see that the y-values approach the same value whether we move from the left side or from the right side. In other words, we need an input \(x\) within the interval \(6.9 When \(x=7\), there is no corresponding output. We write it as lim x → af(x) = L Here is the other table, which approaches the x-value 2 from the right. A sequence is one type of function, but functions that are not sequences can also have limits. Numerical Approach to Limits Example 1: Let f (x) = 2 x + 2 and compute f (x) as x takes values closer to 1. As we look at this table, we certainly see symmetry. Here is the problem we were working on the last section. The coordinate pair of the point would be \((a,f(a)).\) If such a point exists, then \(f(a)\) has a value. We will continue this process within the next section. In this section, we will examine numerical and graphical approaches to identifying limits.\[1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8}... \nonumber \]gets closer and closer to 0. If the left- and right-hand limits are equal, we say that the function \(f(x)\) has a two-sided limit as \(x \) approaches \(a.\) More commonly, we simply refer to a The limit of a function \(f(x)\), as \(x\) approaches \(a\), is equal to \(L\), that is, \[ \lim_{x \to a} f(x)=L \nonumber \] if and only if \[\lim_{x \to a^−} f(x)= \lim_{x \to a^+} f(x). Both \(a\) and \(L\) must be real numbers. To avoid changing the function when we simplify, we set the same condition, \(x≠7\), for the simplified function. Note that the value of the limit is not affected by the output value of \(f(x)\) at \(a\). We write it asFor the following limit, define \(a,f(x),\) and \(L.\)First, we recognize the notation of a limit. Centering around \(x=0,\) we choose two viewing windows such that the second one is zoomed in closer to \(x=0\) than the first one. This is so because the function is undefined at the x-value 2. Therefore, the limit does not exist. ...we would have to approach the x-value -5 from the left. There are two ways to demonstrate Calculus limits: a numerical approach or a graphical approach. Let us also gather a table for this limit, which is the other side (right side) of the x-value -5. The notationindicates that as the input \(x\) approaches 7 from either the left or the right, the output approaches 8. \nonumber \]\[ \lim_{x \to 0^+} \left( 3 \sin \left( \dfrac{π}{x} \right) \right) \;\;\; \text{does not exist.} We can represent the function graphically as shown in Figure \(\PageIndex{2}\).What happens at \(x=7\) is completely different from what happens at points close to \(x=7\) on either side. Leaving the calculation to the reader, we gain these values for the finding the limit as we approach the x-value 2 from the left. \nonumber \]Numerically estimate the following limit: \( \lim \limits_{x \to 0} (\sin (\frac{2}{x}))\).Access these online resources for additional instruction and practice with finding limits.\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)12.2: Finding Limits - Numerical and Graphical Approaches[ "article:topic", "limits", "right-hand limits", "left-hand limits", "two-sided limits", "license:ccby", "showtoc:no", "authorname:openstaxjabramson" ][ "article:topic", "limits", "right-hand limits", "left-hand limits", "two-sided limits", "license:ccby", "showtoc:no", "authorname:openstaxjabramson" ]\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)Principal Lecturer (School of Mathematical and Statistical Sciences)Example \(\PageIndex{1}\): Understanding the Limit of a FunctionHOW TO: Given a function \(f(x)\), use a graph to find the limits and a function value as \(x\) approaches \(a.\)Example \(\PageIndex{2}\): Finding a Limit Using a GraphHow to: Given a function \(f\),use a table to find the limit as \(x\) approaches \(a\) and the value of \(f(a)\), if it exists.Example \(\PageIndex{3}\): Finding a Limit Using a TableQ & A: Is it possible to check our answer using a graphing utility?Example \(\PageIndex{4}\): Using a Graphing Utility to Determine a Limit \nonumber \]The values of \(f(x)\) can get as close to the limit \(L\) as we like by taking values of \(x\) sufficiently close to \(a\) but greater than \(a\). Knowledge of these two characteristics will help you create accurate tables for numerical analysis when calculating limits. Note: this table is the result of calculating within radians. We first consider values of x approaching 1 from the left (x < 1). Numerical and graphical approaches are used to introduce to the concept of limits using examples. This may be phrased with the equation \( \lim_{x \to 2}(3x+5)=11,\) which means that as \(x\) nears 2 (but is not exactly 2), the output of the function \(f(x)=3x+5\) gets as close as we want to \(3(2)+5,\) or \(11\), which is the limit \(L\), as we take values of \(x\) sufficiently near 2 but not at \(x=2\).For the following limit, define \(a,f(x)\), and \(L\).We can approach the input of a function from either side of a value—from the left or the right. limits functions table of values numerical approach undefined If the function has a limit as \(x\) approaches 0, state it. This is why we are using this numerical approach. We write the solution like so: Hence, 4 is the limit. The y-values fluctuate without headed toward any particular value.
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