1. Limits and Continuity

Determine the following limits.

$\underset{\theta \to 0}{\lim }\frac{2+\sin \theta }{1-{\cos }^2\theta }$

Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is

f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0

where ln x has zero mean and standard deviation σ > 0.

b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

b. lim x→−2 f(x)

Evaluate ${\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}}$ and${\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}$.

$f\left(x\right)=\frac{6e^x+20}{3e^x+4}$

Explain the meaning of lim x→a f(x) =∞.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x² − 4x + 3) / (x − 1)

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

15. lim(x→∞)arctan(x)

Determine the following limits. 

lim x→−∞ (x+ √x^2−5x)

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.

d. lim x→0^+ f(x)

Determine the following limits.

$\underset{x\to 1^{+}}{\lim }\frac{x^2-5x+6}{x-1}$

A right circular cylinder with a height of 10 cm and a surface area of S cm² has a radius given by r(S)=1/2(√100+2S/π −10).

Find lim S→0^+ r(S) and interpret your result.

Evaluate lim x→1 (x^3+3x^2−3x+1).

Use Theorem 3.10 to evaluate the following limits.

lim x🠂2 (sin (x-2)) / (x² - 4)

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim x/(x² − 1) as

d. x→−1⁻

2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

g. e^(cos(x))