1. Limits and Continuity

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

lim t→5 (1/t^2 − 4t − 5 −1/ 6(t − 5))

Let f(x) = {x^2+1 / if x<−1

√x+1 if x≥−1.

Compute the following limits or state that they do not exist.

limx→−1 f(x)

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.

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

lim x→1 x^2 − 1 / x − 1

Determine the following limits. 

lim x→∞ x^4+7 / x^5+x^2−x

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)