1. Limits and Continuity

Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = x²(4x² − √(16x⁴ + 1))

Given the function $f\left(x\right)$ defined as follows:

$f\left(x\right)=\left\{\begin{array}{ll}2x+1& ,x<1\\ 3& ,x=1\\ x^2& ,x>1\end{array}\right.$

Find $\underset{x\to 1}{lim}f\left(x\right)$. (If an answer does not exist, enter 'dne.')

Limits and Infinity

Find the limits in Exercises 37–46.

2x² + 3

lim -------------

x→⁻∞ 5x² + 7

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim (2 − 3 / t¹/³) as

a. t → 0⁺

Limits and Continuity

Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.

e. x + ƒ(x)

Determine the following limits.

lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).

lim x→0 f(x)

Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.

lim x→(−π/2)⁺ sec x

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^+ h(x)

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→−1 (2x − 1)^2 − 9 / x + 1

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (x^4−1)/(x^2−1)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

a. x² + √x

Find the exact length of the curve given by $x=2\sqrt{{\sqrt{t}}^3}$, $y=t^2-2$ for $0\le t\le 4$.

Evaluate each limit. 

lim θ→0 (1/(2+sinθ)-1/2)/sin θ