Limits to Infinity

LIMITS OF FUNCTIONS AS X APPROACHES INFINITY

The following problems require the algebraic computation of limits of functions as x approaches plus or minus infinity. Most problems are average. A few are somewhat challenging. All of the solutions are given WITHOUT the use of L'Hopital's Rule. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by giving careful consideration to the forms

and

during the computations of these limits. Initially, many students INCORRECTLY conclude that

is equal to 1 , or that the limit does not exist, or is

. Many also conclude that

is equal to 0 . In fact, the forms

and

are examples of indeterminate forms. This simply means that you have not yet determined an answer. Usually, these indeterminate forms can be circumvented by using algebraic manipulation. Such tools as algebraic simplification and conjugates can easily be used to circumvent the forms

and

so that the limit can be calculated.

PROBLEM 1 : Compute .
Click HERE to see a detailed solution to problem 1.
PROBLEM 2 : Compute .
Click HERE to see a detailed solution to problem 2.
PROBLEM 3 : Compute .
Click HERE to see a detailed solution to problem 3.
PROBLEM 4 : Compute .
Click HERE to see a detailed solution to problem 4.
PROBLEM 5 : Compute .
Click HERE to see a detailed solution to problem 5.
PROBLEM 6 : Compute .
Click HERE to see a detailed solution to problem 6.
PROBLEM 7 : Compute .
Click HERE to see a detailed solution to problem 7.
PROBLEM 8 : Compute .
Click HERE to see a detailed solution to problem 8.
PROBLEM 9 : Compute .
Click HERE to see a detailed solution to problem 9.
PROBLEM 10 : Compute .
Click HERE to see a detailed solution to problem 10.
PROBLEM 11 : Compute .
Click HERE to see a detailed solution to problem 11.
PROBLEM 12 : Compute .
Click HERE to see a detailed solution to problem 12.
PROBLEM 13 : Compute .
Click HERE to see a detailed solution to problem 13.
PROBLEM 14 : Compute .
Click HERE to see a detailed solution to problem 14.
PROBLEM 15 : Compute .
Click HERE to see a detailed solution to problem 15.
PROBLEM 16 : Compute limit (x to -infinity) cos [ x/(x^2+10) + pi/3 ] .
Click HERE to see a detailed solution to problem 16.
PROBLEM 17 : Compute .
Click HERE to see a detailed solution to problem 17.
PROBLEM 18 : Compute .
Click HERE to see a detailed solution to problem 18.
PROBLEM 19 : Compute .
Click HERE to see a detailed solution to problem 19.