In Section 2.1.4 we relied on graphical and numerical approaches to investigate limits and estimate their values. In the next four sections we go beyond this intuitive approach and develop tools for computing limits in a precise way. The next theorem provides our first set of tools.
The proof of Theorem 1 is discussed in Section 2.9 and Appendix D. To illustrate the underlying idea, consider two numbers such as 2.99 and 5.001. Observe that 2.99 is close to 3 and 5.0001 is close to 5, so certainly the sum 2.99 + 5.0001 is close to 3 + 5 and the product (2.99)(5.0001) is close to (3)(5). In the same way, if f(x) approaches L and g(x) approaches M as x → c, then f(x) + g(x) approaches the sum L + M, and f(x)g(x) approaches the product LM. The other laws are similar.
If and exist, then
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Before proceeding to the examples, we make some useful remarks.
Use the Basic Limit Laws to evaluate:
Solution
Evaluate (a) and (b).
Solution
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The next example reminds us that the Basic Limit Laws apply only when the limits of both f(x) and g(x) exist.
Show that the Product Law cannot be applied to if f(x) = x and g(x) = x^{−1}.
Solution For all x ≠ 0 we have f(x)g(x) = x · x^{−1} = 1, so the limit of the product exists:
However, does not exist because g(x) = x^{−1} approaches ∞ as x → 0+ and it approaches −∞ as x → 0−. Therefore, the Product Law cannot be applied and its conclusion does not hold:
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