How to Solve Special Cases of Calculus Limit Problems
What are Special Cases of Limit Problems?
A limit is a function involving a variable such as x, y, or k where the function is defined for all values of x except at the point where x approaches a certain value called a. In graphing the limit as x approaches a, the graph is continuous but at the value a, the graph takes on a different value causing a change in the graph. The graph may be undefined at this point.
Limit problems that when solved by direct substitution of a value a, result in 0/0 are special cases of limits that are undefined, but solvable if they are simplified.
A function involving a radical in the denominator is a special case of limits:
lim k =
For examples of how to solve limits involving a difference with radicals in the denominator or numerator see examples 11 and 12 in special cases of calculus limit problems.
Special Cases Involving Difference of Squares and Cubes
Sometimes a limit problem will involve a difference of squares. If direct substitution results in 0/0, the limit should be simplified by multiplying the function by the conjugate (the factor with the addition of squares).
To review the factors in a difference of squares see: PurpleMath Special Factoring: Difference of Squares.
If a limit problem involves the difference of cubes, and direct substitution results in 0/0, the limit should be expanded or factored into its special form of: a3 – b3 = (a – b)(a2 + ab + b2)
More examples of factoring sums and difference of cubes can be found at: PurpleMath Special Factoring.
Students should also practice other advanced algebra concepts such as being able to put a factored form of an equation back into the form of a difference of cubes. This will help in solving special cases of limits which have to be rewritten.
An example of a limit involving a difference of cubes in the denominator is:
lim x =
For examples of how to solve limits involving a difference with difference of squares and cubes in numerator and denominator see examples 13 and 14 in special cases of calculus limit problems.
More Special Cases of Limits
Sometimes limits are in a form which needs to be rewritten into its factored form.
An example of a limit that can be rewritten is:
x→-2 x + 2
To solve a limit that direct substitution results in ∞/∞, divide by the highest power of x.
An example of a limit that needs to be divided by x2 is:
Lim 2x2+ 5=
Note that when x2 is under the radical sign, it is expressed as x4
For examples of how to solve limits of the above two forms, see examples 15 and 16 in special caes of calculus limit problems.
The special cases involving limits of different forms need to be simplified in different ways to be solved. Students will recognize these special cases in limit problems and will be able to solve them using this study guide.