Infinite Limits and Limits at Infinity

Question

Exercise 4.2 (Number 1 and 2)

1.)

lim

-> 4+ ( -(x-2) / (4-x) + (5x) / (2x+3) )

2.)

lim

-> 3- ( (2x/x-3) - (x+8)/(5x+2) )

*note: please include also a solution and a step-by-step explanation for Exercise 4.2 (Number 1 And 2), so I can understand deeply each problem.*

Exercise 4.3 (Number 1 to 3)

1.)

lim

x-> +∞ ( (8x^4-2x^3+x^2+7x-1) / (2x^4+2x^3-3x+11)

2.)

lim

x-> +∞ ( (7x^2+2x-10) / (2x^3-x^2+9x) )

3.)

lim

x-> -∞ ( (-3x^2+12x-1) / (2x+3) )

*note: please include also a solution and a step-by-by explanation for Exercise 4.3 (Number 1 to 3), so I can understand deeply each problem.

THANK YOU SO MUCH IN ADVANCE! It's my rusty at the moment in my Basic Calculus Subject.

Answer 1

4.2.1) There are asymptotes at x=4 and x = -3/2. If you plot the function, approaching the limit from the right cause -(x-2)/(4-x) to become -∞, which swamps out the rest to the equation. Note, approaching from the left (lim x→4^-) would be ∞.

4.2.2) There is an asymptote at x=3, so we only are concerned with -(x+2)/(x-3), lim x→3^- = ∞ and

lim x→3⁺ = -∞

4.3.1 For limit problems like this, when the highest degrees in the numerator and the denominator are the same, the problem is reduced to

lim x→+∞ 8x⁴/(2x⁴) = 8/2 = 4

Or you can multiply the numerator and denominator by 1/x⁴ and using lim x→+∞ 1/xⁿ = 0, you get the same answer.

4.3.2 Considering only the highest degrees of the numerator and denominator polynomials, this can be be reduced to

lim x→+∞ 7x²/(2x³) = lim x→+∞ 7/2x = 0

4.3.3 For the same reasons as above, this can be reduced to

lim x→-∞ -3x²/(2x) = lim x→-∞ -3x/2 = -(-∞) = ∞