Two Limit to Infinity Definition

Two Limits of Functions as X Approaches Infinity:

The problems need the algebraic calculation of limits of functions as x approaches + or – endless. The majority problems are average.

The minority are rather challenging.

Initially, many students incorrectly conclude that oo/oo is equal to 1, or that the limit does not exist, or is + oo or -oo . Many also conclude that oo - oo  is equal to 0.

In fact, the forms oo /oo and oo - oo are examples of indeterminate forms.

Such tools as algebraic generalization and conjugates can easily be used to avoid the forms oo / oo and oo - oo so that the limit can be calculated.


Two limit to infinity problems:

Two limit to infinity problem 1:

Compute

lim x ->oo 100/ (x^2 + 5)

SOLUTION 1:

lim x -> oo 100/(x^2 +5) = 100/oo = 0.

(The top element is always 100 and the base element x2+5approaches oo as x approachesoo , so that the resulting fraction approaches 0.)

Two limit to infinity problem 2:

Compute

lim x -> -oo 7/ (x^3 - 20).

SOLUTION 2 :

lim x -gt -oo 7/(x^3 -20) = 7 / -oo = 0.

(The top element is always 7 and the base element x^3 - 7 approaches -oo as x approaches -oo , so that the resulting fraction approaches 0.)

Two limit to infinity problem 3:

Compute

lim x -> oo 3x^3 - 1000 x^ 2.

SOLUTION 3 :

lim x -gt oo ( 3x^3 - 1000x^2) = oo - oo

(This is NOT equal to 0. It is an undefined form.)

= lim x -gt oo x^2(3x - 1000)

(As x approaches oo , each of the two expressions x^2 and 3 x - 1000 approaches oo .)

= (oo) (oo)

(This is NOT an indeterminate form. It has meaning.)

=oo.


(Therefore, the limit does not exist. Note that an alternate solution follows by first factoring out x^3, the highest power of x . Try it.)

Two limit to infinity practice problems:

Compute

lim x -> -oo x^4+5x^2+1.

Compute

lim x-> oo x^5-x^2+x-10.