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At this point, you should have a very strong intuitive sense of what the limit of a function means and how you can find it. In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language.
We now demonstrate how to use the epsilon-delta definition of a limit to construct a rigorous proof of one of the limit laws. The triangle inequality is used at a key point of the proof, so we first review this key property of absolute value.
We’ll be looking at the precise definition of limits at finite points that have finite values, limits that are infinity and limits at infinity. We’ll also give the precise, mathematical definition of continuity.
At this point, you should have a very strong intuitive sense of what the limit of a function means and how you can find it. In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language.
formal definition of a limit. Definition: The Limit Suppose f(x) is defined on an open interval about x 0, not necessarily containing x 0. We say that L is the limit of f(x) as x approaches x 0, written lim x→x 0 f(x) = L if for every number > 0, there exists a corresponding number δ > 0 such that for all x with 0 < |x−x 0| < δ we have ...
Learning Objectives. Describe the epsilon-delta definition of a limit. Apply the epsilon-delta definition to find the limit of a function. Describe the epsilon-delta definitions of one-sided limits and infinite limits. Use the epsilon-delta definition to prove the limit laws.
This section introduces the formal definition of a limit. Many refer to this as "the epsilon--delta,'' definition, referring to the letters ϵ and δ of the Greek alphabet. Before we give the actual definition, let's consider a few informal ways of describing a limit.
The Precise Definition of a Limit. Learning Objectives. Describe the epsilon-delta definition of a limit. Apply the epsilon-delta definition to find the limit of a function. Describe the epsilon-delta definitions of one-sided limits and infinite limits. Use the epsilon-delta definition to prove the limit laws.
The Precise Definition of a Limit. Learning Objectives. Describe the epsilon-delta definition of a limit. Apply the epsilon-delta definition to find the limit of a function. Use the epsilon-delta definition to prove the limit laws. Use the epsilon-delta definition to find deltas algebraically.
When we’re evaluating a limit, we’re looking at the function as it approaches a specific point.