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Let a be a real number. In this case, we find the limit by performing addition and then applying one of our previous strategies. Since from the squeeze theorem, we obtain. Then, each of the following statements holds: Sum law for limits: Difference law for limits: Constant multiple law for limits: Product law for limits: Quotient law for limits: for. The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions. Evaluating a Limit When the Limit Laws Do Not Apply. The following observation allows us to evaluate many limits of this type: If for all over some open interval containing a, then. For evaluate each of the following limits: Figure 2. 19, we look at simplifying a complex fraction. Limits of Polynomial and Rational Functions. Simple modifications in the limit laws allow us to apply them to one-sided limits. Find an expression for the area of the n-sided polygon in terms of r and θ.

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By now you have probably noticed that, in each of the previous examples, it has been the case that 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. Problem-Solving Strategy: Calculating a Limit When has the Indeterminate Form 0/0. 3Evaluate the limit of a function by factoring. We now take a look at a limit that plays an important role in later chapters—namely, To evaluate this limit, we use the unit circle in Figure 2.
20 does not fall neatly into any of the patterns established in the previous examples. Factoring and canceling is a good strategy: Step 2. In the first step, we multiply by the conjugate so that we can use a trigonometric identity to convert the cosine in the numerator to a sine: Therefore, (2. Step 1. has the form at 1. Evaluate each of the following limits, if possible. The next examples demonstrate the use of this Problem-Solving Strategy. 4Use the limit laws to evaluate the limit of a polynomial or rational function. If the numerator or denominator contains a difference involving a square root, we should try multiplying the numerator and denominator by the conjugate of the expression involving the square root. Consequently, the magnitude of becomes infinite. 24The graphs of and are identical for all Their limits at 1 are equal. Use radians, not degrees.

Then we cancel: Step 4. First, we need to make sure that our function has the appropriate form and cannot be evaluated immediately using the limit laws. Using the expressions that you obtained in step 1, express the area of the isosceles triangle in terms of θ and r. (Substitute for in your expression. Let's apply the limit laws one step at a time to be sure we understand how they work. Since is the only part of the denominator that is zero when 2 is substituted, we then separate from the rest of the function: Step 3. and Therefore, the product of and has a limit of. The function is undefined for In fact, if we substitute 3 into the function we get which is undefined. In the figure, we see that is the y-coordinate on the unit circle and it corresponds to the line segment shown in blue. 31 in terms of and r. Figure 2. To see this, carry out the following steps: Express the height h and the base b of the isosceles triangle in Figure 2. Next, using the identity for we see that. 27 illustrates this idea. Equivalently, we have.

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Think of the regular polygon as being made up of n triangles. We now take a look at the limit laws, the individual properties of limits. Use the limit laws to evaluate. 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.

17 illustrates the factor-and-cancel technique; Example 2. He never came up with the idea of a limit, but we can use this idea to see what his geometric constructions could have predicted about the limit. 28The graphs of and are shown around the point. 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. T] The density of an object is given by its mass divided by its volume: Use a calculator to plot the volume as a function of density assuming you are examining something of mass 8 kg (. Since for all x in replace in the limit with and apply the limit laws: Since and we conclude that does not exist. Where L is a real number, then. Let and be defined for all over an open interval containing a. After substituting in we see that this limit has the form That is, as x approaches 2 from the left, the numerator approaches −1; and the denominator approaches 0. Evaluating a Limit by Factoring and Canceling. The graphs of and are shown in Figure 2. The first two limit laws were stated in Two Important Limits and we repeat them here. The next theorem, called the squeeze theorem, proves very useful for establishing basic trigonometric limits.

We see that the length of the side opposite angle θ in this new triangle is Thus, we see that for. We now use the squeeze theorem to tackle several very important limits. Because for all x, we have. Because and by using the squeeze theorem we conclude that. To find this limit, we need to apply the limit laws several times. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. Evaluating a Limit by Simplifying a Complex Fraction. The radian measure of angle θ is the length of the arc it subtends on the unit circle. Let's begin by multiplying by the conjugate of on the numerator and denominator: Step 2.

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By taking the limit as the vertex angle of these triangles goes to zero, you can obtain the area of the circle. It now follows from the quotient law that if and are polynomials for which then. The function is defined over the interval Since this function is not defined to the left of 3, we cannot apply the limit laws to compute In fact, since is undefined to the left of 3, does not exist. Then, To see that this theorem holds, consider the polynomial By applying the sum, constant multiple, and power laws, we end up with. Do not multiply the denominators because we want to be able to cancel the factor. 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. Both and fail to have a limit at zero. 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.
These two results, together with the limit laws, serve as a foundation for calculating many limits. In this section, we establish laws for calculating limits and learn how to apply these laws. Power law for limits: for every positive integer n. Root law for limits: for all L if n is odd and for if n is even and. 30The sine and tangent functions are shown as lines on the unit circle.
25 we use this limit to establish This limit also proves useful in later chapters. 6Evaluate the limit of a function by using the squeeze theorem. Use the squeeze theorem to evaluate. Additional Limit Evaluation Techniques. Let and be polynomial functions. Now we factor out −1 from the numerator: Step 5. Assume that L and M are real numbers such that and Let c be a constant.

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