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Based on Example 2.6, we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.
- Chapter 2
5.1 Approximating Areas; 5.2 The Definite Integral; 5.3 The...
- Chapter 2
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. In this section, we establish laws for calculating limits and learn how to apply these laws.
2.3.1 Recognize the basic limit laws. 2.3.2 Use the limit laws to evaluate the limit of a function. 2.3.3 Evaluate the limit of a function by factoring. 2.3.4 Use the limit laws to evaluate the limit of a polynomial or rational function. 2.3.5 Evaluate the limit of a function by factoring or by using conjugates.
You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. The Squeeze Theorem allows you to find the limit of a function if the function is always greater than one function and less than another function with limits that are known.
Limit Rules. Limit of a constant \lim_ {x\to {a}} {c}=c. Basic Limit \lim_ {x\to {a}} {x}=a. Squeeze Theorem. \mathrm {Let\:f,\:g\:and\:h\:be\:functions\:such\:that\:for\:all}\:x\in [a,b]\:\mathrm { (except\:possibly\:at\:the\:limit\:point\:c),} f (x)\le {h (x)}\le {g (x)}
21 Δεκ 2020 · Evaluate the \(\displaystyle \lim_{x→3}\frac{2x^2−3x+1}{5x+4}\). Solution. Since 3 is in the domain of the rational function \(f(x)=\frac{2x^2−3x+1}{5x+4}\), we can calculate the limit by substituting 3 for x into the function. Thus, \[\lim_{x→3}\frac{2x^2−3x+1}{5x+4}=\frac{10}{19}. \nonumber\]
In the overwhelming cases of real applications we only have to worry about limits when the function involves division by 0. For example f(x) = (x4+x2+1)=x needs to be investigated more carefully at x = 0. You see for example that for x = 1=1000, the function is slightly larger than 1000.
