Understanding Limits in Algebra
Limits describe the value that a function approaches as the input approaches a certain point. They are foundational in calculus and analysis.
What is a Limit?
A limit is the value a function approaches as the input gets closer to a specific point.
It is written as \( \lim_{x \to a} f(x) \).
Basic Limit Properties
- \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \)
- \( \lim_{x \to a} c = c \)
One-Sided and Infinite Limits
One-sided limits approach from the left or right.
Infinite limits describe behavior as \( x \) approaches infinity.
Applications of Limits
Limits are used to define derivatives, integrals, and continuity.
Practice Problems
- Find \( \lim_{x \to 2} (x^2 - 4) \).
- Evaluate \( \lim_{x \to 0} \frac{\sin x}{x} \).
- Find the limit as \( x \to \infty \) of \( \frac{1}{x} \).
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