Two-sided limits using advanced algebra

The  exercise appears under the Precalculus Math Mission and Differential calculus Math Mission. This exercise explores calculating limits with some advanced algebraic manipulation.

Types of Problems
There is one type of problem in this exercise:


 * 1) Find the limit using algebra: This problem has a limit of a function approaching a number. The student is expected to find the limit and write the correct answer in the space provided.Tslua1.png

Strategies
Knowledge of the evaluating functions, factoring polynomials, rationalization and fractions are encouraged to ensure success on this exercise.
 * 1) The first method for solving a limit is to plug in the limiting value for x. If this is defined, the result is the answer.
 * 2) The second method for solving limits on this exercise is to factor the numerator and denominator, reduce, and then plug in. The result will be the answer.
 * 3) If there is a radical in the limit, multiply by the conjugate in either the numerator or denominator as appropriate to eliminate the radical.
 * 4) If there are fraction in the limit, combine fractions with the complex fraction techniques before to assist simplification.
 * 5) L'hopital's rule comes later in the standard calculus sequence, but after introduced it can be used to perform this exercise more efficiently.
 * 6) A calculator can be used by evaluating the function at a number "very close" to the limiting value.

Real-life Applications

 * 1) Limits are the foundation for both differential and integral calculus.
 * 2) The concept of continuity is rigorously defined via limits.
 * 3) Asymptotes of rational functions can be understood more rigorously by looking at them as limits.