AP 微积分关于Limit极限的题型总结带答案.docx

1、AP 微积分关于Limit极限的题型总结带答案1、 isTo solve this problem, you need to remember how to evaluate limits. Always do limit problems on the first pass. Whenever we have a limit of a polynomial fraction wherex, we divide the numerator and the denominator, separately, by the highest power ofxin the fraction. Now

2、take the limit. Remember that the, ifn 0, wherekis a constant. Thus, we get2、If we take the limit asxgoes to 0, we get an indeterminate form, so lets use LHpitals Rule. We take the derivative of the numerator and the denominator and we get:Now, when we take the limit we get:.3、There are two importan

3、t trigonometric limits to memorize.The first step that we always take when evaluating the limit of a trigonometric function is to rearrange the function so that it looks like some combination of the limits above. We can do this by factoring sinxout of the numerator.Now we can break this into limits

4、that we can easily evaluate. Now if we take the limit asx 0 we get 4(1)(0) = 0.4、This mayappearto be a limit problem, but it isactuallytesting to see whether you know the definition of the derivative.:You should recall that the definition of the derivative saysThus, if we replacef(x) with tan (x), w

5、e can rewrite the problem asThe derivative of tanxis sec2x. Thus, 5、We take the limit asxgoes to 0, we get an indeterminate form, so lets use LHpitals Rule. We take the derivative of the numerator and the denominator and we get .When we take the limit, we again get an indeterminate form, so lets use

6、 LHpitals Rule a second time. We take the derivative of the numerator and the denominator and we get.Now, when we take the limit we get:.6、Notice that if we plug 5 into the expressions in the numerator and the denominator, we get, which is undefined. Before we give up, we need to see if we can simpl

7、ify the limit so that it can be evaluated. If we factor the expression in the numerator, we getwhich can be simplified tox+ 5.Now, if we take the limit (by plugging in 5 forx), we get 10.Notice that if we plug 5 into the expressions in the numerator and the denominator, we get, which is undefined. B

8、efore we give up, we need to see if we can simplify the limit so that it can be evaluated. If we factor the expression in the numerator, we getwhich can be simplified tox+5.Now, if we take the limit (by plugging in 5 forx), we get 10.7、EvaluateNotice how this limit takes the form of the definition o

9、f the derivative, which isHere, if we think off(x) as 5x4, then this expression gives the derivative of 5x4at the pointx=.The derivative of 5x4isf(x)=20x3.Atx=, we get.8、Findkso thatis continuousfor allx.In order forf(x) to be continuous at a pointc,there are three conditions that need to be fulfill

10、ed.(1)f(c) exists. (2)exists. (3)First, lets check condition (1):f(4) exists; its equal tok.Next, lets check condition (2).From the left side, we getFrom the right side, we getTherefore, the limit exists, and.Now, lets check condition (3). In order for this condition to be fulfilled,kmust equal 8.9、

11、If we take the limit asxgoes to 0, we get an indeterminate form, so lets use LHpitals Rule. We take the derivative of the numerator and the denominator and we get.Now, when we take the limit we get:. 10、First, rewrite the limit asNext, break the fraction intoNow, if we multiply the top and bottom of

12、 the first fraction by 8, we getNow, we can take the limit, which gives us 8(1)(1) = 8.11、If we take the limit asxgoes to 0, we get an indeterminate form, so lets use LHpitals Rule. We take the derivative of the numerator and the denominator and we get. When we take the limit, we again get an indete

13、rminate form, so lets use LHpitals Rule a second time. We take the derivative of the numerator and the denominator and we get.Now, when we take the limit we get.12、If we take the limit asxgoes to , we get an indeterminate form, so lets use LHpitals Rule. We take the derivative of the numerator and t

14、he denominator and we get.When we take the limit, we again get an indeterminate form, so lets use LHpitals Rule a second time. We take the derivative of the numerator and the denominator and we get13、First, rewrite the limit asNext, break the expression into two rational expressions.This can be brok

15、en up further intoWe will evaluate the limit of each separately.First expression.Divide the top and bottom byx:Then, multiply the top and bottom of the upper expression by 3, and the top and bottom of the lower expression by 5.Now, if we take the limit, we get Second expression .This limit is straig

16、htforward:.Third expressionFirst, pull the constant, 3, out of the limit:.Now, if we multiply the top and bottom of the expression by 5, we getNow, if we take the limit, we get.Combine the three numbers, and we get.14、Evaluate.Notice how this limit takes the form of the definition of the derivative,

17、 which isIf we think off(x) as sinx, then this expression gives the derivative of sinxat the point.The derivative of sinxisf(x) = cosx. At, we get15、Use the double angle formula for sine, sin2=2sincos, to rewrite the limit and then solve:16、If we take the limit asxgoes to, we get an indeterminate fo

18、rm, so lets use LHpitals Rule. We take the dervative of the numerator and the denominator and we get.Now, when we take the limit we get17、What is ？Recall the definition of the derivative says:and the derivative of secxis tanxsecx.Thus,. Therefore, the limit does not exist.18、If we take the limit asx

19、goes to 0 from the right, we get an indeterminate form, so lets use LHpitals Rule. We take the derivative of the numerator and the denominator and we get. Now, when we take the limit we get.19、Either use LHpitals rule or recall that . In this case,can be rewritten as20、21、Find.When you insert forx,

20、the limit is, which is indeterminate. First, rewrite the limit as. Then, use LHpitals Rule to evaluate the limit:. This limit exists and equals 0.22、Notice the limit is in the form of the definition of the derivative. You could evaluate the limit, but if you see the definition of the derivative and

21、the main function,f(x) = 2x2, it is easier to evaluate the derivative directly. Thus, the solution isf(x) = 4x.23、Find.Use LHpitals Rule.24、If we take the limit asxgoes tofrom the left, we get an indeterminate form, so lets use LHpitals Rule. We take the derivative of the numerator and the denominat

22、or and we getWe can simplify this using trig identities Now, when we take the limit we get25、Use the Rational Function Theorem.26、27、Since |x- 2|=2 -xifx 2, the limit as28、Note thatwheref(x) = lnx. 29、Nonexistent30、(where x is the greatest integer inx) isNonexistent，Here,and .31、Divide both numerato

23、r and denominator by32、 The given limit equals f(x) = sinx.33、Nonexistent34、Since the degrees of numerator and denominator are the same, the limit asx is the ratio of the coefficients of the terms of highest degree:35、is an indeterminate form of the type. ApplyingLHpitalsRule, you have.36、The diagra

24、m above shows the graph of a functionf for -2 x4. Which of the following statements is/are true?I.existsII.existsIII.existsA. I only B. II only C. I and II only D. I, II, and IIIThe correct answer is (C).Examining the graph, note that.Since the two one-sided limits are equal,exists. Statement I is t

25、rue. Also, note that .Therefore, statement II is true, but statement III is false because the two one-sided limits are not the same.37、Note that asx approaches-. . . .38、Note that the definition of the derivative. .39、, .Therefore,is an indeterminate form of. ApplyingLHpitalsRule, you have.40、What is the, if