1. Limits and Continuity

Determine the following limits. 

lim x→−∞ 40x^4+x^2+5x / √64x^8+x^6

Determine the following limits. 

lim x→∞ 6x²/(4x^2+√(16x⁴ + x²))

Determine the following limits. 

lim x→−∞ (x+ √x^2−5x)

Determine the following limits. 

lim x→∞ sin x / e^x

Determine the following limits. 

lim x→−∞ (e^x cos x +3)

Describe the end behavior of g(x) = e^-2x.

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)