Finding Limits
For the function f whose graph is given, determine the following limits. Write ∞ or −∞ where appropriate.
h. lim x → ∞ f(x)
Finding Limits
For the function f whose graph is given, determine the following limits. Write ∞ or −∞ where appropriate.
h. lim x → ∞ f(x)
[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.
a. How does the graph behave as x → 0⁺?
Give reasons for your answers.
y = (3/2)(x − (1 / x))²/³
[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.
c. How does the graph behave near x = 1 and x = −1?
Give reasons for your answers.
y = (3/2)(x − (1 / x))²/³
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (tan 2x) / x
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 (1 − cos θ) / sin 2θ
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
llimx→0 (x −x cos x) / sin² 3x
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limh→0 sin(sin h) / sin h
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 sin θ cot 2θ
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 tan θ / θ²cot 3θ
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limx→0 (cos²x − cos x) / x²
Graphing Simple Rational Functions
Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.
y = (x + 3)/(x + 2)
Graphing Simple Rational Functions
Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.
y = 2x/(x + 1)
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = 2x / (x² − 1)
Domains and Asymptotes
Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.
y = (√(x² + 4)) / x
Finding Limits of Differences When x → ±∞
Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)
lim x → −∞ (2x + √(4x² + 3x − 2))