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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) = 2x - 3| x - 3
Find the limit (if it exists). If it does not exist, explain why.lim x→2− (2[[x]] + 1)
Find the limit (if it exists). If it does not exist, explain why. lim √x(x - 1) X-1+
Find the limit (if it exists). If it does not exist, explain why.lim x→4 [[x − 1]]
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) -2x, x ≤ 2 x² - 4x + 1, x > 2
Find the limit L. Then use the - definition to prove that the limit is L.lim x→6 3
Discuss the continuity of the function on the closed interval. h(x) 3 5-x' [0, 5]
Find the one-sided limit (if it exists).lim x→(1/2)− x sec x
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) = Jesc / . 6 2, ²x - 3| ≤2 |x - 3| > 2
Find the limit L. Then use the - definition to prove that the limit is L.lim x→2 (-1)
Find the limit L. Then use the - definition to prove that the limit is L.lim x→0 3√x
Find the limit L. Then use the - definition to prove that the limit is L.lim x→4 √x
Find the limit L. Then use the - definition to prove that the limit is L.lim x→-5 |x - 5|
Find the limit L. Then use the - definition to prove that the limit is L.lim x→3 |x - 3|
Use the information to determine the limits.lim x→c f(x) = -∞lim x→c g(x) = 3(a) lim x→c [f(x) + g(x)](b) lim x→c [f(x)g(x)](c) lim x→c g(x)/f(x)
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = x2 − x + 20
Find the limit L. Then use the - definition to prove that the limit is L.lim x→ 1 (x2 + 1)
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = tan x/2
Find the limit L. Then use the - definition to prove that the limit is L.lim x→-4 (x2 + 4x)
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) = R2² 1 9
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) x + 3 x² - 3x - 18
Find the limit of the trigonometric function. lim x-0 3(1- cos x) X
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = 5 − [[x]]
Find the limit of the trigonometric function. lim x-0 (sin x)(1 cos x) x²
Find the limit of the trigonometric function. lim -0 cos tan
The definition of limit on page requires that f is a function defined on an open interval containing c, except possibly at c. Why is this requirement necessary?
If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain your reasoning.
Use the Intermediate Value Theorem to show thatf(x) = 2x3 − 3has a zero in the interval [1, 2].
Find the limit of the trigonometric function. lim x-0 tan² x X
Find the limit of the trigonometric function. (1 lim h-0 cos h)² h
Use the Intermediate Value Theorem to show thatf(x) = x2 + x − 2has at least two zeros in the interval [−3, 3].
Find the limit of the trigonometric function. lim x-0 6 - 6 cos x 9 3
Use the graph of f to identify the values of c for which lim x→c f(x) exists.(a)(b) -2 6 4 -2 + 2 4
A sporting goods manufacturer designs a golf ball having a volume of 2.48 cubic inches.(a) What is the radius of the golf ball?(b) The volume of the golf ball varies between 2.45 cubic inches and
Consider the functionf (x) = ∣x + 1∣ − ∣x − 1∣ /x.Estimatelim x→0 ∣x + 1∣ − ∣x − 1∣/xby evaluating f at x-values near 0. Sketch the graph of f.
Discuss the continuity of the composite function h(x) = f(g(x)).f(x) = sinxg (x) = x2
Find the limit of the trigonometric function.lim → sec
Determine whether f(x) approaches ∞ or −∞ as x approaches 6 from the left and from the right. = - 1 (x - 6)²
Describe the relationship between precalculus and calculus. List three precalculus concepts and their corresponding calculus counterparts.
Use the graph to find the limit (if it exists). If the limit does not exist, explain why.(a) lim x→2 h(x)(b) lim x→1 h(x) h(x) 3 نیا 2 -1 y H₁ + x² 1 2 3 X
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. x-4 x-4x²5x+4 lim X f(x) 3.9 3.99 3.999 4 ? 4.001 4.01 4.1
Determine whether f(x) approaches ∞ or −∞ as x approaches −2 from the left and from the right. X f(x) = tan - 4 6 32 2 y + 2 1
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. sin x lim x-0 X X f(x) -0.1 -0.01 -0.001 0 0.001 2 0.01 0.1
Given the limit lim x→2 (2x + 1) = 5 use a sketch to show the meaning of the phrase “ 0 < ∣x − 2∣ < 0.25 implies ∣(2x + 1) − 5∣ < 0.5.”
Create a table of values for the function and use the result to determine whether f (x) approaches ∞ or −∞ as x approaches −3 from the left and from the right. Use a graphing utility to graph
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. lim X √x +1 -1 X -0.1 -0.01 -0.001 0 ? 0.001 0.01 0.1
Find the distance traveled in 15 seconds by an object traveling at a constant velocity of 20 feet per second.
Determine whether f(x) approaches ∞ or −∞ as x approaches 4 from the left and from the right.f(x) = 1 x − 4
Create a table of values for the function and use the result to determine whether f (x) approaches ∞ or −∞ as x approaches −3 from the left and from the right. Use a graphing utility to graph
Find the limit L. Then use the - definition to prove that the limit is L.lim x→1 (x + 4)
Find the limit L. Then use the - definition to prove that the limit is L.lim x→2 (1 - x2)
Consider the length of the graph of f(x) = 5/x from (1, 5) to (5, 1).(a) Approximate the length of the curve by finding the distance between its two endpoints, as shown in the first figure.(b)
Create a table of values for the function and use the result to determine whether f (x) approaches ∞ or −∞ as x approaches −3 from the left and from the right. Use a graphing utility to graph
Find the limit (if it exists). If it does not exist, explain why. X lim x--3- 2-9
Find the vertical asymptotes (if any) of the graph of the function. f(x) 3 x² + x - 2
Find the vertical asymptotes (if any) of the graph of the function. f(x) = x² x² - 4
Create a table of values for the function and use the result to explain why the limit does not exist. 2 lim x=0x³
Find the limit.lim t→4 √t + 2
Find the vertical asymptotes (if any) of the graph of the function. f(x) x²2x15 1³ nở - 5x + x − 5 -
Find the vertical asymptotes (if any) of the graph of the function. f(x) = 4x² + 4x - 24 x42x³9x² + 18x
Find the vertical asymptotes (if any) of the graph of the function.f(x) = 1/x2
Find the vertical asymptotes (if any) of the graph of the function. s(t) = t sin t
Find the limits.f(x) = 5 − x, g(x) = x3(a) lim x→1 f(x)(b) lim x→4 g(x)(c) lim x→1 g( f(x))
Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not, explain why.(a) f(1)(b) lim x→1 f(x)(c) f(4)(d) lim x→4 f(x)
Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. x³ + 729 lim x-9 x +9
Find the limit (if it exists). If it does not exist, explain why.lim x→ cot x
Find the limits.f(x) = 4 − x2, g(x) = √x+1(a) lim x→1 f(x)(b) lim x→3 g(x)(c) lim x→1 g( f(x))
Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. lim x-0 √√2x + 9-3
Determine whether the graph of the function has a vertical asymptote or a removable discontinuity at x = −1. Graph the function using a graphing utility to confirm your answer. f(x) = = cos(x² -
Find the vertical asymptotes (if any) of the graph of the function.f(x) = csc x
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) = 1 4x 2
Find the limit (if it exists). If it does not exist, explain why. lim h(t), where h(t) 1³ + 1, t
Sketch a graph of a function f that satisfies the given values. f(0) is undefined.lim x→0 f(x) = 4 f(2) = 6lim x→2 f(x) = 3
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) x² 2 X
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? X tan f(x) = 4 x₂ |x < 1 |x ≥ 1
Find the limit L. Then find such that ∣f(x) − L∣ < whenever 0 < ∣x − c∣ < for(a) = 0.01 and(b) = 0.005.lim x→2 (3x + 2)
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) = x+1, x≤2 3-x, x > 2
Use the information to evaluate the limits.lim x→c f(x) = 16(a) lim x→c [ f(x)]2(b) lim x→c √f(x)(c) lim x→c [3f (x)](d) lim x→c [ f(x)]3/2
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = 6/x
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = 3x − cos x
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) = 4 X-5
Discuss the continuity of the function on the closed interval.g(x) = √8 − x3 , [−2, 2]
Use the information to determine the limits.lim x→c f(x) = ∞lim x→c g(x) = −2(a) lim x→c [f(x) + g(x)](b) lim x→c [f(x)g(x)](c) lim x→c g(x)/f(x)
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = x4 − 81x
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = csc 2x
Use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = [[x − 8]]
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? f(x) = 3 X 1³- X
Determine whether f(x) approaches ∞ or −∞ as x approaches 6 from the left and from the right. f(x) 1 X-6
For a long-distance phone call, a hotel charges $9.99 for the first minute and $0.79 for each additional minute or fraction thereof. A formula for the cost is given byC(t) = 9.99 − 0.79[[1 − t]],
Describe the intervals on which the function is continuous.f(x) = √x + cos x
When using the definition of limit to prove that L is the limit of f(x) as x approaches c, you find the largest satisfactory value of . Why would any smaller positive value of also work?
Discuss the continuity of the composite function h(x) = f(g(x)).f(x) = x2g (x) = x - 1
If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2? Explain your reasoning.
Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.f(x) = x2 + 5x − 4, [−1, 2], f(c) = 2
Find functions f and g such that lim x→c f(x) = ∞ and lim x→c g(x) = ∞, but lim x→c [ f(x) − g(x)] ≠ 0.
Consider the functionf(x) = (1 + x)1/x.Estimatelim x→0 (1 + x)1/xby evaluating f at x-values near 0. Sketch the graph of f.
Find the vertical asymptotes (if any) of the graph of the function. f(x) = sec X 2
Use a graphing utility to graph the function. Use the graph to determine any x-values at which the function is not continuous.f(x) = [[x]] − x
Prove that if lim x→c f(x) = ∞, then lim x→c 1 f(x) = 0.
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