1. Limits and Continuity

Determine the following limits. 

lim θ→∞ cos θ / θ²

Determine the following limits. 

lim t→∞ (5t² + t sin t) / t²

Determine the following limits. 

lim x→−∞ (5 + 100/x + sin⁴ x³ / x²)

Determine the following limits. 

lim x→∞ (3x¹² − 9x⁷)

Determine the following limits. 

lim x→−∞ (3x⁷ + x²) 

Determine the following limits. 

lim x→−∞ (2x^-8 + 4x³)

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

f(x) = (x² − 4x + 3) / (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) = (2x³ + 10x² + 12x) / (x³ + 2x²)

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

f(x) = (√(16x⁴ + 64x²) + x²) / (2x² − 4) 

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

f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

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

f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

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))

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

f(x) = (x² − 9)/(x(x−3))

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

f(x) = (x⁴ − 1)/(x^2−1)