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study help
mathematics
precalculus
Questions and Answers of
Precalculus
Let P(3, 4) be a point on the circle x2 + y2 = 25 (see figure).(a) What is the slope of the line joining P and O(0, 0)?(b) Find an equation of the tangent line to the circle at P.(c) Let Q(x, y) be
What is meant by an indeterminate form?
Determine whether f(x) approaches ∞ or −∞ as x approaches −2 from the left and from the right. f(x) 1 x + 2 ليا 3 2 -1 -2 -3 y X
Discuss the relationship between secant lines through a fixed point and a corresponding tangent line at that fixed point.
In your own words, describe what is meant by a vertical asymptote of a graph.
List the two special trigonometric limits.
Is the limit of f(x) as x approaches c always equal to f(c)? Why or why not?
Consider the graphs of the four functions g1, g2, g3, and g4.For each given condition of the function f, which of the graphs could be the graph of f ?(a) lim x→2 f(x) = 3(b) f is continuous at
Find the distance traveled in 15 seconds by an object moving with a velocity of v(t) = 20 + 7 cos t feet per second.
Find the limit.lim x→−3 x4
Find the values of the constants a and b such thatlim x→0 √a + bx − √3 x = √3.
Find the limit.lim x→−3 (2x + 5)
Find the limit L. Then use the - definition to prove that the limit is L.lim x→9 √x
Find the limit.lim x→9 (4x − 1)
Find the limit.lim x→−3 (x2 + 3x)
Find the limit L. Then use the - definition to prove that the limit is L.lim x→5 9
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.lim x→2 (−x3 + 1)
To escape Earth’s gravitational field, a rocket must be launched with an initial velocity called the escape velocity. A rocket launched from the surface of Earth has velocity v (in miles per
Sketch the graph of the functionf(x) = ⟨1 x⟩.(a) Evaluate f( 1 4), f(3), and f(1).(b) Evaluate the limits lim x→1− f(x), lim x→1+ f(x), lim x→0− f(x), and lim x→0+ f(x).(c) Discuss
Find the limit (if it exists). If it does not exist, explain why. 2 lim x-3+ x + 3
Find the limit.lim x→−6 x2
Create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. lim x-4x² x + 4 + 9x + 20
Find the limit.lim x→−3 (2x2 + 4x + 1)
Sketch the graph of the function f(x) = ⟨x⟩ + ⟨−x⟩.(a) Evaluate f(1), f(0), f( 1 2), and f(−2.7).(b) Evaluate the limits lim x→1− f(x), lim x→1+ f(x), and lim x→12 f(x).(c)
Find the limit (if it exists). If it does not exist, explain why. 4- x x²16 lim x-4+x2²
Find the limit (if it exists). If it does not exist, explain why. x-5 lim x-5+ x² - 25 2 X-
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.lim x→0 (5x − 3)
Find the limit.lim x→1 (2x3 − 6x + 5)
Find the limit.lim x→2 √x3 + 1
Let a be a nonzero constant. Prove that if lim x→0 f(x) = L, then lim x→0 f(ax) = L. Show by means of an example that a must be nonzero.
Find the limit.lim x→27 (3√x − 1)4
Find the limit.lim x→−4 (1 − x)3
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-0 X
Find the vertical asymptotes (if any) of the graph of the function. f(x) = 2 (x - 3)³
Find the limit (if it exists). If it does not exist, explain why. |x - 10] lim x-10 x 10
Find the limit (if it exists). If it does not exist, explain why. lim Δ.x - 0+ (x + 4x)2 + x + Δx - (x2 + x) Δε
Create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. tan.x lin x-0 tan 2x
Find the limit.lim x→7 (x − 4)3
Find the vertical asymptotes (if any) of the graph of the function. f(x) 3x x² + 9
Find the limit.lim x→0 (3x − 2)4
Create a table of values for the function and use the result to explain why the limit does not exist. /X/C lim x-0 x²
Find the vertical asymptotes (if any) of the graph of the function. h(s) = 3s + 4 s² - 16
Find the limit (if it exists). If it does not exist, explain why. lim f(x), where f(x) x-31 [x² - 4x + 6, -x² + 4x2, x < 3 x ≥ 3
Find the vertical asymptotes (if any) of the graph of the function. g(x) r – 5x + 25 13 x³ + 125
Find the vertical asymptotes (if any) of the graph of the function. h(x) = x² - 9 x³ + 3x²-x-3
Find the limits.f(x) = x+7, g(x) = x2(a) lim x→-3 f(x)(b) lim x→4 g(x)(c) lim x→-3 g( f(x))
Find the vertical asymptotes (if any) of the graph of the function. h(t) 1² - 2t 14 - 16
Find the limit (if it exists). If it does not exist, explain why.lim x→ /2 sec x
Find the limit (if it exists). If it does not exist, explain why. lim x-1 X +3)
Use the graph to find the limit (if it exists). If the limit does not exist, explain why. lim tan x x- /2 디스 2 1 y 2 . 3ㅍ 2 -x
Find the limits.f(x) = 2x2 − 3x + 1, g(x) = 3√x+6(a) lim x→4 f(x)(b) lim x→21 g(x)(c) lim x→4 g( f(x))
Evaluate the limit given lim x→c f(x) = −6 and lim x→c g(x) = 1/2.lim x→c [f(x)g(x)]
Find the limit of the trigonometric function.lim x→ 1cos x/3
Find the vertical asymptotes (if any) of the graph of the function.f(x) = tan x
Evaluate the limit given lim x→c f(x) = −6 and lim x→c g(x) = 1/2.lim x→c f(x)/g(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(−2)(b) lim x→−2 f(x)(c) f(0)(d) lim x→0 f(x)(e)
Find the limit of the trigonometric function.lim x→2 sin x/12
Evaluate the limit given lim x→c f(x) = −6 and lim x→c g(x) = 1/2.lim x→c [f(x) + 2g(x)]
Find the vertical asymptotes (if any) of the graph of the function. 8( tan
The graph of f(x) = x + 1 is shown in the figure. Find such that if 0 < ∣x − 2∣ < , then ∣ f (x) − 3∣ < 0.4. 3.4 2.6 5 ++ 4 3 2 y 0.5 1.0 1.5 2.0 2.5 3.0 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 [1/(x + 4)] (1/4) 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 cos x 1 X -
Evaluate the limit given lim x→c f(x) = −6 and lim x→c g(x) = 1/2.lim x→c [f(x)]2
Find the limit of the trigonometric function.lim x→ cos 3x
Sketch a graph of a function f that satisfies the given values. (There are many correct answers.)f(−2) = 0f(2) = 0lim x→−2 f(x) = 0lim x→2 f(x) does not exist.
Find the limit of the trigonometric function.lim x→3 tan x/4
Find the one-sided limit (if it exists). lim x-2+ x - 2
Find the one-sided limit (if it exists). x² lim x-2-x² + 4
Find the limit of the trigonometric function.lim x→7 sec x/6
Use the information to evaluate the limits.lim x→c f(x) = 2/5lim x→c g(x) = 2(a) lim x→c [5g(x)](b) lim x→c [ f(x) + g(x)](c) lim x→c [ f(x)g(x)]
Find the one-sided limit (if it exists). x + 3 x--3-x²+x-6 lim
Find the limit (if it exists). If it does not exist, explain why. 1 lim x-3+ x + 3
Find the limit L. Then find such that ∣f(x) − L∣ < whenever 0 < ∣x − c∣ < for (a) = 0.01 and (b) = 0.005. lin x-6 3/
Find the one-sided limit (if it exists). lim x-(-1/2)+ 6x² + x1 4x² - 4x - 3
Find the one-sided limit (if it exists). lim 1 + x-0 x
Find the limit (if it exists). If it does not exist, explain why. 9-x lim x-6x² - 36
Find the limit (if it exists). If it does not exist, explain why. lim x 25 √x-5 x 25
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? 9-x t = (x) f
Find the one-sided limit (if it exists). lim x-14- 1² + 2 x + 4)
Find the one-sided limit (if it exists). (6- lim 6 x-0+ x
Find the limit L. Then find such that ∣f(x) − L∣ < whenever 0 < ∣x − c∣ < for (a) = 0.01 (b) = 0.005.lim x→2 (x2 − 3)
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→4 (x2 + 6)
Find the limit L. Then find such that ∣f(x) − L∣ < whenever 0 < ∣x − c∣ < for (a) = 0.01 (b) = 0.005.lim x→4 (x2 − x)
Find the limit (if it exists). If it does not exist, explain why. lim f(x), where f(x) X-2 (x - 2)², x ≤ 2 2- X, X> 2
Find the one-sided limit (if it exists). ( x − ² / + 3) x limx x-0+
Find the limit (if it exists). If it does not exist, explain why. lim g(x), where g(x): X-1+ √1 - X, x + 1, X ≤ 1 X > 1
Find the one-sided limit (if it exists). lim (sin x + x-0+ 1-) X
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→3 x2
Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?f(x) = sin x − 8x
Find the limit (if it exists). If it does not exist, explain why. lim f(s), where f(s): S-2 -S² - 4s - 2, S² + 4s + 6, S≤-2 S> -2
Find the limit L. Then use the - definition to prove that the limit is L. lim (x - 1) x--4
Find the one-sided limit (if it exists). -2 lim x-( /2)* COS X
Find the one-sided limit (if it exists). x + 2 lim x-0 cotx
Find the one-sided limit (if it exists). lim X- X CSC X
Find the limit L. Then use the - definition to prove that the limit is L.lim x→4 (x + 2)
Find the limit L. Then use the - definition to prove that the limit is L. lim (x + 1) x-3
Find the limit (if it exists). If it does not exist, explain why. lim x-2 ²-4 2 |x - 2|
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