MTH 241 Sections 1.4-1.6
Decide from the graph whether a limit exists. If a limit exists, find its value. 3
Approaches negative infinity from both side, which isn't a number so there is it does not exist
Determine whether limx→2 f(x) exists. If so, give the limit.
As x approaches 2 from the negative side, y=3, as x approaches 2 from the positive side, y=1. These are not equal, therefore the limit doesn't exist.
Is the function graphed to the right continuous at the point x=−1?
At the point x=-1, there is no breaks in the graph, so it is continuous
Is the function graphed at the right differentiable at x=−5?
Derivative does not exist at x=-5
The function below is defined for all x except for one value of x. If possible, define f(x) at the exceptional point in a way that makes f(x) continuous for all x..
Factor the numerator and simplify. Replace value for x and determine value
The function below is defined for all x except for one value of x. If possible, define f(x) at the exceptional point in a way that makes f(x) continuous for all x.
Find derivative then use number in place of x
Decide from the graph whether a limit exists. If a limit exists, find its value. 2
Find y-value when x is 2. The limit is -7
Use limits to compute the derivative. f′(2) where f(x)=x^2-9
First find derivative 2x so f'(2)=2(2)=4
Determine if the following limit exists. If it does exist, compute the limit. x^2+8x/x
If possible, rewrite the limit by simplifying the rational expression. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. Take limits of top and bottom. You see this gets you 0/0, no good, go back and factor the numerator, answer is x+3 Select the correct choice below and fill in any answer boxes in your choice. Plug 0 in for x, gets 3
Determine whether the following limit exists. Compute the limit if it exists.
Is the function f(x)=x2−4x−5/x−5 defined when x=5? Plugin 5 for x, in this case, no Does the function need to be defined at x=5 in order for its limit to exist at x=5? No To evaluate the limit, begin by factoring the numerator.: x2−4x−5. What adds to -4 and multiples to -5? (x-5)(x+1) When f(x) is defined, common binomial factors can be cancelled from the numerator and denominator. Cancel out (x-5) from numerator and denominator to get just (x+1) Now determine the limit of the simplified expression as x approaches 5. Replace x with 5 in (x+1) which is 6.
Decide from the graph whether a limit exists. If a limit exists, find its value.
Limit does not exist in this case
Determine the value of a that makes the function f(x) continuous at x=18.
Plug 18 in for the first equation then set that answer equal to the second, while using x=18. 18^2-144=2a(18), a=5
Compute the following limit if it exists. limx→5 -2x^3+3
Plug 5 in for x, -2(5)^3+3=-247
Compute the following limit if it exists. y^2/7-y
Plug 7 in for y, 7^2/7-7, divide by 0 error, limits DNE
Compute the following limit if it exists. limx→8(7x+8)
Plug 8 in for x, 7(8)+8=64
Is the function below differentiable at x=7?
Recall that in order for f(x) to be differentiable at a, the limit limh→0f(a+h)−f(a)/h must exist. If it is suspected that f(x) is nondifferentiable at a, then it is easier to use the relationship between continuity and differentiability to prove that f(x) is not differentiable at x=a. If f(x) is differentiable at x=a, then what must be true? The function f(x) is continuous at x=a
Compute the following limit if it exists.
Rewrite the limit by using the appropriate definition(s), if possible. Select the correct choice below. Use the constant multiple and the limit of a sum definitions to rewrite the expression as 11lim→1x +limx-->1 8 Select the correct choice below and fill in any answer boxes in your choice. Replace 1 where x is to get 19
Determine if the limit below exists. If it does exist, compute the limit. sqrt x^2-x-6/5-2x
Rewrite the limit using the appropriate limit theorem(s). Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. For fraction, take limit of top and bottom Select the correct choice below and fill in any answer boxes in your choice. Plug in 9 for x to get -sqrt66/13
Determine if the following limit exists. If it does exist, compute the limit. x^5+7x/7
Select the correct choice below and fill in any answer boxes in your choice. Plug in 0 for x, gets 0/0, no go. Factor, x^4+7=7
Determine if the limit below exists. If it does exist, compute the limit. sqrt x^2-3x-10/6-2x
Take limits of top and bottom, plug 10 in for x to get -sqrt60/14
Determine whether the following function is continuous and/or differentiable at x=−5.
The determine if the function f(x)=x^2 is continuous at x=-5: -First condition, find f(-5) and substitute -5 into the f(x)=x^2. f(-5)=(-5)^2=25. So, the function f(x)=x^2 at x=-5 -Second condition, find limx->-5 f(x), (-5)^2=25. So, the limit exists
Is the function, whose graph is drawn to the right, differentiable at x=−3?
There is a sharp corner when x=-3, therefore it is not differentiable
Click the Reset button in the upper right corner. Find limx→2 f(x).
There is a sharp corner, so the function is not continuous
Is the function graphed to the right differentiable at x=−1?
There is a vertical asymptote at x=-1 so it is not differentiable.
Suppose that limx→3f(x)=7 and limx→3g(x)=−5. Find the following limits.
a.limx→3[f(x)+4g(x)] (7+4(-5)=-13 b.limx→3[f(x)g(x)] 7x-5=-35 c.limx→3[3f(x)g(x)] (3)(7)(-5)=-105
Compute the following limit if it exists. limy→4 y^2/4−y
- When assessing limits, if the denominator is heading to 0 and the numerator is not heading to 0, then the limit does not exist, because we are at a vertical asymptote
The owner of a photocopy store charges 9 cents per copy for the first 100 copies and 6 cents per copy for each copy exceeding 100. In addition, there is a setup fee of $1.90 for each photocopying job.
(a) Determine R(x), the revenue from doing one photocopying job consisting of x copies.
The owner of a photocopy store charges 9 cents per copy for the first 100 copies and 6 cents per copy for each copy exceeding 100. In addition, there is a setup fee of $1.80 for each photocopying job.
(a)Determine R(x), the revenue from doing one photocopying job consisting of x copies. If doing 100 copies or less, then R(x)=0.09x+$1.80 If doing more than 100 copies, then R(x)=0.06x+$4.80 (b) If it costs the store owner 4 cents per copy, what is the profit from doing one photocopying job consisting of x copies? (Recall that profit is revenue minus cost.) Subtract 0.04x from each equation. If doing 100 copies or less, 0.05x+$1.80 If doing more than 100 copies, 0.02x+&4.80`
The graph to the right shows the total sales in thousands of dollars for a department store during a typical 24-hour period.
a) Calculate the rate of sales during the period between 6 P.M. and 8 P.M. Use the ARC formula; (24-19)/(8-6)=2.5 b) Which 2-hour interval in the day sees the highest rate of sales and what is the rate? Look for the steepest slope
