Back
Problem 2.4.28
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limt→0 2t / tan t
Problem 2.6.7
Finding Limits
In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)
h(x) = (−5 + (7/x))/(3 – (1/x²))
Problem 2.6.93
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 0 (−1 / x²) = −∞
Problem 2.4.36
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limh→0 sin(sin h) / sin h
Problem 2.4.35
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limt→0 sin(1 − cos t) / (1 − cos t)
Problem 2.1.6
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
P(θ)=θ³ − 4θ² + 5θ; [1,2]
Problem 2.6.94
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 0 (1 / |x|) = ∞
Problem 2.2.13
Calculating Limits
Find the limits in Exercises 11–22.
limt→6 8(t−5)(t−7)
Problem 2.4.16
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
limh→0− (√6 − √(5h² + 11h + 6))/ h
Problem 2.3.40
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
lim x→0 √(4 − x) = 2
Problem 2.1.22
[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.
a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.
b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.
Problem 2.2.58
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = x², x = -2
Problem 2.2.8
Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.
Problem 2.2.38
Limits of quotients
Find the limits in Exercises 23–42.
limx→−1 (√(x² + 8) − 3) / (x + 1)
Problem 2.6.91
Using the Formal Definitions
Use the formal definitions of limits as x → ±∞ to establish the limits in Exercises 91 and 92.
If f has the constant value f(x) = k, then lim x → ∞ f(x) = k.
Problem 2.4.32
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (x² − x + sin x) / 2x
Problem 2.4.37
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 sin θ / sin 2θ
Problem 2.6.44
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0 (−1) / (x² (x + 1))
Problem 2.6.35
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x→∞ (x − 3) / √(4x² + 25)
Problem 2.2.33
Limits of quotients
Find the limits in Exercises 23–42.
limu→1 (u⁴ − 1)/(u³ − 1)
Problem 2.5.13
At what points are the functions in Exercises 13–30 continuous?
y = 1/(x – 2) – 3x
Problem 2.3.45
Using the Formal Definition
Prove the limit statements in Exercises 37–50.
limx→−3 (x² − 9) / (x + 3) = −6
Problem 2.6.90
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → ∞ (√(x² + x) − √(x² − x))
Problem 2.6.23
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x → ∞ √((8x² − 3) / (2x² + x))
Problem 2.6.82
Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.
Problem 2.6.95
Use formal definitions to prove the limit statements in Exercises 93–96.
lim x → 3 (−2 / (x − 3)²) = −∞
Problem 2.5.8
Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
<IMAGE>
At what values of x is f continuous?
Problem 2.6.33
Limits as x → ∞ or x → −∞
The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.
lim x→∞ √(x² + 1) / (x + 1)
Problem 2.1.18
Slope of a Curve at a Point
In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.
y=√7−x, P(−2,3)
Problem 2.6.3
Finding Limits
In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)
f(x) = 2/x − 3