Pre-Calculus Prep: Calculating a Limit Algebraically

In pre-calculus, calculating a limit algebraically involves using algebraic techniques to find the value that a function approaches as the input approaches a specific point. This can be done by applying various limit laws and algebraic manipulations.

Here are the general steps to calculate a limit algebraically:

1. Substitute the value: Try substituting the specific value that the input approaches into the function. If the function is defined at that point, the limit will be equal to the value of the function at that point.

2. Simplify the expression: If substitution doesn't yield a result, try simplifying the expression algebraically. Use algebraic techniques such as factoring, expanding, simplifying fractions, or applying trigonometric identities to manipulate the expression.

3. Apply limit laws: Use limit laws to simplify the expression further. Some common limit laws include the limit of a sum, limit of a product, limit of a quotient, limit of a power, and the limit of a composite function. These laws allow you to break down complex expressions and apply limits to individual components.

4. Evaluate the limit: Once you have simplified the expression as much as possible, try evaluating the limit again. If the expression is in a determinate form (such as 0/0 or ∞/∞), you can often evaluate the limit by canceling out common factors or applying techniques such as L'Hôpital's rule.

5. Use special limits or identities: In some cases, specific limits or identities can help simplify the expression further. For example, trigonometric limits like sin(x)/x as x approaches 0 or exponential limits like e^x as x approaches ∞ have well-known values.

6. Repeat the steps if necessary: If the limit is still not determined or if you encounter indeterminate forms such as 0 * ∞ or ∞ - ∞, you may need to go back to earlier steps and apply further algebraic manipulations or use more advanced techniques like series expansions or logarithmic transformations.

By following these steps and using algebraic techniques, you can calculate limits algebraically and determine the behavior of a function as the input approaches a particular value.

These videos are designed to review and reteach Precalculus and Collegeboard Pre-CALC AP content. My videos cover functions, polynomials, exponential and logarithmic expressions, trigonometry, parametric equations, polar coordinates, vectors, matrices and systems, conic sections, discrete mathematics, sequences and series; and an introduction to calculus.

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa .
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