Find the number of iguanas present in the island intially

Question

Question:

Please ensure that all working is shown and all steps are shown with instructions.

Limits & Continuity

Question 1: Find the limits of the following

(a) Lim_x→∞ (x⁵ - 5x³ + 6x²)/(2x⁶ + 6x⁴ + x³)

(b) Lim_x→0 (5x^1/2 + 3x^2/3 + 4x^3/4)/(6x^1/3 + 9x^1/5)

Question 2:

The population of iguanas introduced into an isolated island is given by

P(t) = 72/(9-t), 0 ≤ t ≤ 9 where t is measured in months.

i. Find the number of iguanas present in the island intially.

ii. Show that the population of iguanas is increasing without bound.

Question 3:

a. Given f(x) =  -1      if x < -2

                     x+1   if -2 ≤ x ≤ 1

                     x² + x + 1  if x > 1

Is f(x) continuous

i. As x tends to -2?

ii. As x tends to 1?

iii. Everywhere throughout its domain?