# limit laws examples and solutions

Example: Solution: We can’t find the limit by substituting x = 1 because is undefined. Let $$f(x),g(x)$$, and $$h(x)$$ be defined for all $$x≠a$$ over an open interval containing $$a$$. Both $$1/x$$ and $$5/x(x−5)$$ fail to have a limit at zero. Step 1. Evaluate the limit of a function by factoring. To find a formula for the area of the circle, find the limit of the expression in step 4 as $$θ$$ goes to zero. Using the Limit Laws, we can write: =\left(\lim_{x→2^−}\dfrac{x−3}{x}\right)\cdot\left(\lim_{x→2^−}\dfrac{1}{x−2}\right). Although this discussion is somewhat lengthy, these limits prove invaluable for the development of the material in both the next section and the next chapter. We can estimate the area of a circle by computing the area of an inscribed regular polygon. \begin{align*} . § Solution f is a polynomial function with implied domain Dom()f = . & = 4 (\blue{-2}) - \red{3}&& \blue{Identity}\hspace{2mm}and\hspace{2mm}\red{Constant}\hspace{2mm}Laws\\ & = -\frac{17} 2 We now practice applying these limit laws to evaluate a limit. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied. \nonumber. This is not always true, but it does hold for all polynomials for any choice of $$a$$ and for all rational functions at all values of $$a$$ for which the rational function is defined. Since neither of the two functions has a limit at zero, we cannot apply the sum law for limits; we must use a different strategy. Dec 22, 2020. This theorem allows us to calculate limits by “squeezing” a function, with a limit at a point a that is unknown, between two functions having a common known limit at $$a$$. & = 0 Use the limit laws to evaluate the limit of a polynomial or rational function. With the first 8 Limit Laws, we can now find limits of any rational function. Use LIMITS OF POLYNOMIAL AND RATIONAL FUNCTIONS as reference. Since $$x−2$$ is the only part of the denominator that is zero when 2 is substituted, we then separate $$1/(x−2)$$ from the rest of the function: $=\lim_{x→2^−}\dfrac{x−3}{x}⋅\dfrac{1}{x−2} \nonumber$. \\ \end{align*} \displaystyle\lim_{x\to -7} (\blue{x} + \red{5}) & = \blue{\lim_{x\to -7} x} + \red{\lim_{x\to-7} 5} && \mbox{Addition Law}\\ The Central Limit Theorem illustrates the Law of Large Numbers. Also, assume $$\displaystyle\lim\limits_{x\to a} f(x)$$ and $$\displaystyle\lim\limits_{x\to a} g(x)$$ both exist. $$, Suppose$$f(x) \geq g(x)$$for all$$x$$near$$x = a$$.$$. 2.3.2 Use the limit laws to evaluate the limit of a function. Interactive simulation the most controversial math riddle ever! Choice (d) is correct! & = 5^3\\ Make use of it. 2.3.5 Evaluate the limit of a function by factoring or by using conjugates. Evaluate $$\displaystyle\lim_{x→3}\dfrac{x^2−3x}{2x^2−5x−3}$$. For polynomials and rational functions, $\lim_{x→a}f(x)=f(a).$. & = \left(\blue{\lim_{x\to 5} x}\right)\left(\red{\lim_{x\to5} x}\right)&& \mbox{Multiplication Law}\\ Evaluate $$\displaystyle \lim_{x→−3}\dfrac{x^2+4x+3}{x^2−9}$$. To see this, carry out the following steps: 1.Express the height $$h$$ and the base $$b$$ of the isosceles triangle in Figure $$\PageIndex{6}$$ in terms of $$θ$$ and $$r$$. Solution. The following observation allows us to evaluate many limits of this type: If for all $$x≠a,\;f(x)=g(x)$$ over some open interval containing $$a$$, then, $\displaystyle\lim_{x→a}f(x)=\lim_{x→a}g(x).$. We now take a look at a limit that plays an important role in later chapters—namely, $$\displaystyle \lim_{θ→0}\dfrac{\sin θ}{θ}$$. The Greek mathematician Archimedes (ca. You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. The limit of a constant is that constant: $$\displaystyle \lim_{x→2}5=5$$. Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. Problem-Solving Strategy: Calculating a Limit When $$f(x)/g(x)$$ has the Indeterminate Form $$0/0$$. % Evaluate limit lim x→∞ 1 x As variable x gets larger, 1/x gets smaller because 1 is being divided by a laaaaaaaarge number: x = 1010, 1 x = 1 1010 The limit is 0. lim x→∞ 1 x = 0. Use the method in Example $$\PageIndex{8B}$$ to evaluate the limit. $\lim_{x→a}x=a \quad \quad \lim_{x→a}c=c \nonumber$, $\lim_{θ→0}\dfrac{\sin θ}{θ}=1 \nonumber$, $\lim_{θ→0}\dfrac{1−\cos θ}{θ}=0 \nonumber$. Step 6. We now take a look at the limit laws, the individual properties of limits. The following three examples demonstrate the use of these limit laws in the evaluation of limits. \end{align*} Since 3 is in the domain of the rational function $$f(x)=\displaystyle \frac{2x^2−3x+1}{5x+4}$$, we can calculate the limit by substituting 3 for $$x$$ into the function. Legal. & & \text{Apply the basic limit results and simplify.} (1) Constant Law: $$\displaystyle\lim\limits_{x\to a} k = k$$, (2) Identity Law: $$\displaystyle\lim\limits_{x\to a} x = a$$, (3) large Addition Law: $$\displaystyle\lim\limits_{x\to a} f(x) + g(x) = \displaystyle\lim\limits_{x\to a} f(x) + \displaystyle\lim\limits_{x\to a} g(x)$$, (4) Subtraction Law: $$\displaystyle\lim\limits_{x\to a} f(x) - g(x) = \displaystyle\lim\limits_{x\to a} f(x) - \displaystyle\lim\limits_{x\to a} g(x)$$, (5) Constant Coefficient Law: $$\displaystyle\lim\limits_{x\to a} k\cdot f(x) = k\displaystyle\lim\limits_{x\to a} f(x)$$, (6) Multiplication Law: $$\lim\limits_{x\to a} f(x)\cdot g(x) = \left(\lim\limits_{x\to a} f(x)\right)\left(\lim\limits_{x\to a} g(x)\right)$$, (7) Power Law: $$\displaystyle\lim\limits_{x\to a} \left(f(x)\right)^n= \left(\displaystyle\lim\limits_{x\to a} f(x)\right)^n$$ provided $$\displaystyle\lim\limits_{x\to a} f(x)\neq 0$$ if $$n <0$$, (8) Division Law: $$\displaystyle\lim\limits_{x\to a} \frac{f(x)}{g(x)} = \frac{\displaystyle\lim\limits_{x\to a}f(x)}{\displaystyle\lim\limits_{x\to a} g(x)}$$ provided $$\displaystyle\lim\limits_{x\to a} g(x)\neq 0$$. Work with each term separately, then subtract the results. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Thus, since $$\displaystyle \lim_{θ→0^+}\sin θ=0$$ and $$\displaystyle \lim_{θ→0^−}\sin θ=0$$, Next, using the identity $$\cos θ=\sqrt{1−\sin^2θ}$$ for $$−\dfrac{π}{2}<θ<\dfrac{π}{2}$$, we see that, $\lim_{θ→0}\cos θ=\lim_{θ→0}\sqrt{1−\sin^2θ}=1.\nonumber$. & = \sin(\blue\pi) && \mbox{Identity Law}\\ \lim_{x\to 3} (8x) & = 8\,\lim_{x\to 3} x && \mbox{Constant Coefficient Law}\\ We then need to find a function that is equal to $$h(x)=f(x)/g(x)$$ for all $$x≠a$$ over some interval containing a. Graph $$f(x)=\begin{cases}−x−2, & \mathrm{if} \; x<−1\\ 2, & \mathrm{if} \; x=−1 \\ x^3, & \mathrm{if} \; x>−1\end{cases}$$ and evaluate $$\displaystyle \lim_{x→−1^−}f(x)$$. If the exponent is negative, then the limit of the function can't be zero! & =-11 Step 4. Central Limit Theorem for the Mean and Sum Examples. Then $$\lim\limits_{x\to a} f\left(g(x)\right) = f\left(\lim\limits_{x\to a} g(x)\right) = f(M)$$. & = 3 & = 4\left(\blue{\displaystyle\lim_{x\to-2} x}\right)^3 + 5\,\red{\displaystyle\lim_{x\to-2} x} && \mbox{Power Law}\\ That is, $$f(x)/g(x)$$ has the form $$K/0,K≠0$$ at a. Think of the regular polygon as being made up of $$n$$ triangles. For root functions, we can find the limit of the inside function first, and then apply the root. (In this case, we say that $$f(x)/g(x)$$ has the indeterminate form $$0/0$$.) Then, $\lim_{x→a}\frac{p(x)}{q(x)}=\frac{p(a)}{q(a)}$, To see that this theorem holds, consider the polynomial, $p(x)=c_nx^n+c_{n−1}x^{n−1}+⋯+c_1x+c_0.$, By applying the sum, constant multiple, and power laws, we end up with, \begin{align*} \lim_{x→a}p(x) &= \lim_{x→a}(c_nx^n+c_{n−1}x^{n−1}+⋯+c_1x+c_0) \\[4pt] &= c_n\left(\lim_{x→a}x\right)^n+c_{n−1}\left(\lim_{x→a}x\right)^{n−1}+⋯+c_1\left(\lim_{x→a}x\right)+\lim_{x→a}c_0 \\[4pt] &= c_na^n+c_{n−1}a^{n−1}+⋯+c_1a+c_0 \\[4pt] &= p(a) \end{align*}. $$\dfrac{\dfrac{1}{x+1}−\dfrac{1}{2}}{x−1}$$ has the form $$0/0$$ at 1. \displaystyle\lim_{x\to 5} x^2 & = \displaystyle\lim_{x\to 5} (\blue{x}\cdot \red{x})\\ and the function $$g(x)=x+1$$ are identical for all values of $$x≠1$$. $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, 2.3: Calculating Limits Using the Limit Laws, https://math.libretexts.org/@app/auth/2/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FCalculus%2FMap%253A_Calculus__Early_Transcendentals_(Stewart)%2F02%253A_Limits_and_Derivatives%2F2.03%253A_Calculating_Limits_Using_the_Limit_Laws, $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$. Let $$f(x)$$ and $$g(x)$$ be defined for all $$x≠a$$ over some open interval containing $$a$$. Choice (b) is incorrect . Find an expression for the area of the $$n$$-sided polygon in terms of $$r$$ and $$θ$$. The proofs that these laws hold are omitted here. The limit of a positive integer root of a function is the root of the limit of the function: It is assumed that if is even. b. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Ask yourself, why they were o ered by the instructor. The limit of a constant is that constant: \ (\displaystyle \lim_ {x→2}5=5\). The proofs that these laws hold are omitted here. We substitute (“plug in”) x =1 and evaluate f ()1 . & = e^{-1}\\ In fact, since $$f(x)=\sqrt{x−3}$$ is undefined to the left of 3, $$\displaystyle\lim_{x→3^−}\sqrt{x−3}$$ does not exist. Thus, we see that for $$0<θ<\dfrac{π}{2}$$, $$\sin θ<θ<\tanθ$$. Evaluate $$\displaystyle \lim_{x→−3}\dfrac{\dfrac{1}{x+2}+1}{x+3}$$. & = \blue{\frac 1 2} - \red{9} && \blue{Identity}\hspace{2mm}and\hspace{2mm}\red{Constant}\hspace{2mm}Laws\\ The first two examples are the first two examples considered in Lesson 2; the final example is new. Let’s apply the limit laws one step at a time to be sure we understand how they work. This completes the proof. Because $$\displaystyle \lim_{θ→0^+}0=0$$ and $$\displaystyle \lim_{x→0^+}θ=0$$, by using the squeeze theorem we conclude that. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied. $$. We now turn our attention to evaluating a limit of the form $$\displaystyle \lim_{x→a}\dfrac{f(x)}{g(x)}$$, where $$\displaystyle \lim_{x→a}f(x)=K$$, where $$K≠0$$ and $$\displaystyle \lim_{x→a}g(x)=0$$. Example $$\PageIndex{7}$$: Evaluating a Limit When the Limit Laws Do Not Apply. The graphs of these two functions are shown in Figure $$\PageIndex{1}$$. Evaluate the limit of a function by using the squeeze theorem. Using Limit Laws Repeatedly. Example $$\PageIndex{4}$$ illustrates the factor-and-cancel technique; Example $$\PageIndex{5}$$ shows multiplying by a conjugate. Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. Solution Let In Figure 11.9, you can see that as approaches 0, oscillates between and 1. Evaluate using a table of values. 287−212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. Factoring and canceling is a good strategy: $\lim_{x→3}\dfrac{x^2−3x}{2x^2−5x−3}=\lim_{x→3}\dfrac{x(x−3)}{(x−3)(2x+1)}\nonumber$. Evaluate the $$\displaystyle \lim_{x→3}\frac{2x^2−3x+1}{5x+4}$$. In the figure, we see that $$\sin θ$$ is the $$y$$-coordinate on the unit circle and it corresponds to the line segment shown in blue. b. Evaluate each of the following limits, if possible. Examples of the Central Limit Theorem Law of Large Numbers. Encourage students to investigate limits using a variety of approaches. There is a concise list of the Limit Laws at the. All you have to be able to do is find the limit of each individual function separately. – Typeset by FoilTEX – 8. & = 125 Since $$f(x)=4x−3$$ for all $$x$$ in $$(−∞,2)$$, replace $$f(x)$$ in the limit with $$4x−3$$ and apply the limit laws: $\lim_{x→2^−}f(x)=\lim_{x→2^−}(4x−3)=5\nonumber$. In Mathematics, a limit is defined as a value that a function approaches the output for the given input values. These two results, together with the limit laws, serve as a foundation for calculating many limits. By applying these limit laws we obtain $$\displaystyle\lim_{x→3^+}\sqrt{x−3}=0$$. & = -2 Question 1 Questions What is the value of the limit lim x → 1 x 2 − x − 2 x 2 − 2 x? For $$f(x)=\begin{cases}4x−3, & \mathrm{if} \; x<2 \\ (x−3)^2, & \mathrm{if} \; x≥2\end{cases}$$, evaluate each of the following limits: Figure illustrates the function $$f(x)$$ and aids in our understanding of these limits. Learn more.$$. Use the same technique as Example $$\PageIndex{7}$$. The technique of estimating areas of regions by using polygons is revisited in Introduction to Integration. The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time. 5. \[\begin{align*} \lim_{x→2}\frac{2x^2−3x+1}{x^3+4}&=\frac{\displaystyle \lim_{x→2}(2x^2−3x+1)}{\displaystyle \lim_{x→2}(x^3+4)} & & \text{Apply the quotient law, make sure that }(2)^3+4≠0.\\[4pt] &= \lim_{θ→0}\dfrac{\sin^2θ}{θ(1+\cos θ)}\\[4pt] Because $$−1≤\cos x≤1$$ for all $$x$$, we have $$−x≤x \cos x≤x$$ for $$x≥0$$ and $$−x≥x \cos x ≥ x$$ for $$x≤0$$ (if $$x$$ is negative the direction of the inequalities changes when we multiply). & = 4\,\blue{\lim_{x\to-2} x} - \red{\displaystyle\lim_{x\to-2} 3}&& \mbox{Constant Coefficient Law}\\ Example $$\PageIndex{8A}$$: Evaluating a One-Sided Limit Using the Limit Laws. • With the solution of Example 2 in mind, let’s try to save time by letting (x, y) → (0, 0) along any nonvertical line through the origin. Solution. 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