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In Example 2.22 we look at one-sided limits of a piecewise-defined function and use these limits to draw a conclusion about a two-sided limit of the same function.
- 2.4 Continuity
2.4.1 Explain the three conditions for continuity at a...
- 1.3 Trigonometric Functions
Learning Objectives. 1.3.1 Convert angle measures between...
- 1.2 Basic Classes of Functions
1.2.5 Identify a rational function. 1.2.6 Describe the...
- 3.8 Implicit Differentiation
Find d y d x d y d x for y y defined implicitly by the...
- 5.2 The Definite Integral
5.2.1 State the definition of the definite integral. 5.2.2...
- 3.3 Differentiation Rules
Learning Objectives. 3.3.1 State the constant, constant...
- 3.1 Defining The Derivative
3.1.3 Identify the derivative as the limit of a difference...
- 3.6 The Chain Rule
Learning Objectives. 3.6.1 State the chain rule for the...
- 2.4 Continuity
Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.
Calculus Cheat Sheet Limits. Limits. Definitions Precise Definition : We say lim = = → f ( x ) L if Limit at Infinity : We say lim f x L if we. x →∞. ( ) for every ε > 0 there is a δ > 0 such that can make f ( x ) as close to L as we want by whenever 0 < x − a < δ then f ( x ) − L < ε . taking x large enough and positive. .
17 Αυγ 2024 · Use the limit laws to evaluate the limit of a polynomial or rational function. Evaluate the limit of a function by factoring or by using conjugates. Evaluate the limit of a function by using the squeeze theorem. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values.
Infinite Limit : We say lim f ( x ) = ¥ if we. x a. can make f ( x ) arbitrarily large (and positive) by taking x sufficiently close to a (on either side of a) without letting x = a . There is a similar definition for lim f. x a ( x ) = -¥. except we make f ( x ) arbitrarily large and negative.
2.5.1 Describe the epsilon-delta definition of a limit. 2.5.2 Apply the epsilon-delta definition to find the limit of a function. 2.5.3 Describe the epsilon-delta definitions of one-sided limits and infinite limits. 2.5.4 Use the epsilon-delta definition to prove the limit laws.
Use the limit laws to evaluate the limit of a polynomial or rational function. Evaluate the limit of a function by factoring or by using conjugates. Evaluate the limit of a function by using the squeeze theorem. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values.
