Question 1.

Question

lim h->0(sqrt 49+h-7)/h =

14
1/14***
        
       0
       7
        -1/7

Question 2.  
lim x->infinity(12+x-3x^2)/(x^2-4)=

-3***
       -2
       0
       2
       3

Question 3. 
lim x->infinity (5x^3+x^7)/(e^x)=

infinity***
       0
       -1
        
       3

Question 4. Given that:
x	6.8	6.9	6.99	7.01	7.1	7.2
g(x)	9.44	10.21	10.92	-11.08	-11.31	-12.56

it would appear that lim x->7 g(x)=

0
       7
       11***
       The limit does not exist.
       x + 4

Question 5. Let f be defined as follows, where a ≠ 0, 
f(x)={(x^2-2ax+a^2)/(x-a) if x ≠ a
     {5 if x=a

Which of the following are true about f ?
I. lim f(x) x-> a exists.
II. f(a) exists.
III. f(x) is continuous at x=a.

None
       I, II, and III
       I only
       II only
       I and II only***

Question 7.  lim x-> infinity (2+x-x^2)/(2x+sqrt(4x^4-3)) =

infinity 
       0  
       1***
       4
       -1/2

Question 8.   lim x->0+ (cosx/x)=

0
       1/2
       1
       sqrt(2)/2 
       The limit does not exist.***

Question 9. If g(x) is continuous for all real numbers and g(3) = −1, g(4) = 2, which of the following are necessarily true?
 I. g(x)=1 at least once.
II. lim x->3.5 g(x)=g(3.5)
III. lim x->3- g(x) = lim x->3+ g(x)

I only
       II only
       I and II only
       I, II, and III
       None of these

Question 10. If the following function is continuous, what is the value of a + b?

h(t){3t^2-2t + 1, if t<0
    {acos(t)+b, if 0≤t≤pi/3
    {4sin^2t, if t>pi/3

0
       1
       2
       3
       4

Reiny · Answer

#1 and #2 are correct #3 --- here is a neat trick that works for most limit questions. Use your calculator and try a number close to your approach value e.g. for #1 I used h = .001 and then evaluated the expression for #3, try a "large" number. However for this one even a good calculator overloads even for relative small "large" numbers I tried x = 1000 and got ERROR 1 on my calculator I backed right down to x = 50 , and the result was appr 1.5 x 10^-10 looks very close to zero to me. #4 all values <7 yield positive number getting close to 11 and values >7 are suddenly negative but close to -11 I would say the limit does not exist. skipping #5 #7 intuitive approach ... as x --> infinity, √(4x^4 - 3) ---> 2x^ so we approach (2 + x - x^2)/(2x + 2x^2 which approaching -1/2 (also try my method of checking with a calculator) I used x = 1000 and got -.499 or close to -1/2 #8 correct

Steve · Answer

#3 is 0. exponentials grow much faster than any power of x. l'Hospital's Rule shows that given enough iterations, the derivatives in the numerator go to zero while that pesky e^x remains in the bottom. #4. Note that g(x) changes sign, so the limit does not exist. The limit from the left is different from the limit on the right. #7. As x gets huge, only the highest power matters, so (2+x-x^2)/(2x+sqrt(4x^4-3)) -> -x^2/√4x^4 = -x^2/2x^2 = -1/2 #9. all are true #10. h must be continuous at x=0, so since lim x->0- = 1 acos(0)+b = a+b we must have 1 = a+b

Explain Bot · Answer

Before I answer each individual question, I want to explain how to evaluate limits in general. Evaluating limits involves taking the limit of a function as a variable approaches a certain value. To evaluate a limit, you should follow these steps: 1. First, plug in the approaching value into the function. If you get a number, that is the limit. 2. If you get an indeterminate form, such as 0/0 or infinity/infinity, you need to simplify the function further before evaluating the limit. 3. Simplify the function by using algebraic techniques such as factoring, rationalizing the denominator, or applying properties of limits (e.g., sum, difference, product, or quotient rules). 4. After simplification, plug in the approaching value again and evaluate the limit. Now, let's go through each of the questions and explain how to get the answers: Question 1: The limit is of the form 0/0, so we need to simplify the function. We can start by rationalizing the numerator by multiplying it by the conjugate of the expression inside the square root. After simplification, the function becomes (h)/(h * sqrt(49 + h) + 7). Plugging in h = 0 into this simplified function gives 1/14, so the answer is 1/14. Question 2: The limit can be evaluated by looking at the highest power of x in the numerator and denominator. In this case, it is x^2 in both. Dividing both numerator and denominator by x^2, we get (-3 + 1/x)/(1 - 4/x^2). As x approaches infinity, 1/x goes to 0, so the expression simplifies to -3/-4 = 3/4. Therefore, the answer is 3/4. Question 3: Since x is approaching infinity, the term x^7 becomes much larger than the other terms. Therefore, we can ignore the other terms and consider only (x^7)/(e^x). As x approaches infinity, (x^7) increases much faster than e^x, so the limit is infinity. Question 4: To evaluate this limit, we need to look at the behavior of the function as x approaches 7 from both sides. As x approaches 7 from the left side (values less than 7), the function values increase. However, as x approaches 7 from the right side (values greater than 7), the function values decrease. Therefore, the limit does not exist. Question 5: For this question, we need to consider each statement separately. Statement I: The limit exists because as x approaches a, the function approaches the value of 2a as we can simplify the expression. Statement II: f(a) is defined as 5, so it exists. Statement III: The function is continuous at x = a because both the numerator and the denominator are continuous and f(a) is defined and equals 5. Therefore, statements I and II are true. Question 7: This limit can be simplified by dividing both the numerator and denominator by x. As x approaches infinity, the expression simplifies to (2/x + 1 - x)/(2 + sqrt(4 - 3/x^4)). As x approaches infinity, 2/x goes to 0, so the expression simplifies to 1/2. Therefore, the answer is 1/2. Question 8: Multiplying the numerator and denominator by cos(x) yields (cos(x))^2/x. As x approaches 0, (cos(x))^2 approaches 1, so the limit becomes 1/0, which is undefined. Therefore, the limit does not exist. Question 9: For statement I, we do not have enough information about the behavior of g(x) between x = 3 and x = 4. Therefore, it is not necessarily true. For statement II, g(x) is continuous between x = 3 and x = 4 because it is continuous for all real numbers and g(3) = g(4). For statement III, since g(x) is continuous for all real numbers, the limit from the left and the limit from the right will be equal. Therefore, it is true. Therefore, statements II and III are necessarily true. Question 10: To find the value of a + b, we need to make sure the function is continuous at t = pi/3. By substituting t = pi/3 into the function, we get acos(pi/3) + b = (a/2) + b. Since the function is continuous, the value of h(t) as t approaches pi/3 from both sides should be equal. This means (a/2) + b = 4sin^2(pi/3). Simplifying, we get (a/2) + b = 3. Therefore, the value of a + b is 3. I hope this explanation helps! Let me know if you have any further questions.