![]() ![]() Some texts propose very complicated means to prove this. Decide which is the best of the choices given and indicate your responses in the book.ĭO NOT USE A GRAPHING CALCULATOR FOR ANY OF THESE PROBLEMS.ġ. Each piece of the limit can be done separately and combined at the end.ĭirections: Solve each of the following problems. Therefore, you get:Įxample 11 demonstrates that the presence of addition and subtraction does not affect the outcome of the limit. You are allowed to multiply by 3/3 in Example 11(c) because doing so is the same as multiplying by 1-you are not changing the value of the fraction.īy Special Limit Rule 2, Note that π/x will follow Special Limit Rule 1, since π is a constant. Notice that 3 ∙ 1 is not affected by the limit statement at all in the last step the x may be approaching 0, but there are no x’s left in the problem! However, if you multiply the fraction by 3/3, you get This is not quite the form of Special Limit Rule 3-the 3x and x have to match. The resulting sum will have no limiting value, so the limit is 0 + ∞ = ∞. The last two limits are possible because of Special Limit Rule 1.Īlthough 5/x 3 has a limit of 0 as x approaches infinity, 4x 2 will grow infinitely large. You can do each of these limits separately and add the results: Justification: Again, L’Hopital’s Rule makes this easy, but the graph will suffice as proof also, any value of a will make this true.Įxample 11: Evaluate each of the following limits. Note that the formula above uses a instead of x, because the rule holds true for more than just For example, you can set a = 5x 3 and the formula still works: Justification: This is easily proven by L’Hopital’s Rule, but that is outside the spectrum of Calculus AB, so ABers will have to satisfy themselves with the graph of as proof. For example, if x = 100,000,000, which is approximately the value of e, accurate to seven decimal places. To visually verify that the limit is accurate, use your graphing calculator to calculate for a very large value of x. That small difference becomes magnified when raised to the x power, and the result is e. ![]() Justification: Using the first special limit rule, Technically, is a number very close to, but not quite, one. Consider The denominator is growing larger more quickly than in the previous example, while 7 remains constant. This limit rule works the same way for other eligible n values. The denominator will eventually get so large, in fact, that no matter how large the numerator is, the fraction has an extremely small value so close to zero that the difference is negligible. Remember that the denominator is approaching infinity, so it is getting very large, while the numerator is remaining fixed. Justification: In the simplest case (c = 1, n =1), you are considering You should know the graph of 1/x by heart, and its height clearly approaches zero as x approaches infinity. if c is a nonzero constant and n is a positive integer (BC students have still another topic to cover concerning limits-L’Hopital’s Rule-but that occurs in Chapter 5.) I call these “special” limits because we accept them without formal proof and because of the special way they make you feel all tingly inside.ġ. Before you are completely proficient at limits, however, there are four limits you need to be able to recognize on sight. You have a number of techniques available to you now to evaluate limits and to interpret the continuity that is dependent on those limits. ![]()
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