LC - Evaluating Limits Analytically Lesson

Evaluating Limits Analytically

In previous lessons we have guessed the value of limits based on graphical and numerical evidence, but such approaches do not always lead to the correct conclusion. Instead, finding the value of limits analytically using provable limit laws ensures that a correct answer is found.

Properties of Limits

Suppose that c is a constant and f and g are functions with equation image indicator $LaTeX: \lim_{x\to a}f\left(x\right)=L$ $\lim_{x\to a}f\left(x\right)=L$ and $LaTeX: \lim_{x\to a}g\left(x\right)=K\\$ $\lim_{x\to a}g\left(x\right)=K\\$ . Then

1. Sum: equation image indicator $LaTeX: \lim_{x\to a}[f\left(x\right)+g\left(x\right)]=\lim_{x\to a}f\left(x\right)+\lim_{x\to a}g\left(x\right)=L+K$ $\lim_{x\to a}[f\left(x\right)+g\left(x\right)]=\lim_{x\to a}f\left(x\right)+\lim_{x\to a}g\left(x\right)=L+K$

2. Difference: equation image indicator $LaTeX: \lim_{x\to a}[f\left(x\right)-g\left(x\right)]=\lim_{x\to a}f\left(x\right)-\lim_{x\to a}g\left(x\right)=L-K$ $\lim_{x\to a}[f\left(x\right)-g\left(x\right)]=\lim_{x\to a}f\left(x\right)-\lim_{x\to a}g\left(x\right)=L-K$

3. Product: $LaTeX: \lim_{x\to a}[f\left(x\right)g\left(x\right)]=\lim_{x\to a}f\left(x\right)\cdot \lim_{x\to a}g\left(x\right)=LK$ $\lim_{x\to a}[f\left(x\right)g\left(x\right)]=\lim_{x\to a}f\left(x\right)\cdot \lim_{x\to a}g\left(x\right)=LK$

4. Quotient: $LaTeX: \lim_{x\to a}[\frac{f\left(x\right)}{g\left(x\right)}]=\frac{\lim_{x\to a}f\left(x\right)}{\lim_{x\to a}g\left(x\right)}=\frac{L}{K}$ $\lim_{x\to a}[\frac{f\left(x\right)}{g\left(x\right)}]=\frac{\lim_{x\to a}f\left(x\right)}{\lim_{x\to a}g\left(x\right)}=\frac{L}{K}$ if equation image indicator $LaTeX: \lim_{x\to a}[g\left(x\right)\ne 0$ $\lim_{x\to a}[g\left(x\right)\ne 0$

5. Scalar Multiple: equation image indicator $LaTeX: \lim_{x\to a}[kf\left(x\right)]=k\lim_{x\to a}f\left(x\right)=kL$ $\lim_{x\to a}[kf\left(x\right)]=k\lim_{x\to a}f\left(x\right)=kL$ , where k is a constant

6. Power: equation image indicator $LaTeX: \lim_{x\to a}[f\left(x\right)]^n=L^n$ $\lim_{x\to a}[f\left(x\right)]^n=L^n$ , where n is a positive integer

Three additional special limits are useful:

$LaTeX: \lim_{x\to a}k=k$ $\lim_{x\to a}k=k$ , where k is a constant
$LaTeX: \lim_{x\to a}x=a$ $\lim_{x\to a}x=a$
$LaTeX: \lim_{x\to a}x^n=a^n$ $\lim_{x\to a}x^n=a^n$ , where n is a positive integer

Strategies for Finding Limits Analytically

Finding limits involving algebraic expressions is relatively straightforward since ordinary algebraic operations such as factoring, dividing, rationalizing, etc. are possible. It is only when the expressions aren't algebraic, e.g., trigonometric, or when direct substitution produces a meaningless fractional form 0/0 that more creative approaches are necessary.

Direct Substitution

Using direct substitution for evaluating limits is valid for all polynomial and rational functions with nonzero denominators.

Dividing Out

Simplifying a rational function by factoring and dividing out common factors is very useful in solving many limit problems involving fractional expressions, particularly when the denominator is 0. Recall that the quotient property of limits requires the denominator to be nonzero.

Rationalizing

Another technique for finding limits when direct substitution produces the indeterminate form 0/0 is rationalizing the numerator or the denominator. This technique of multiplying both the numerator and denominator by the conjugate effectively removes the offending expression. View the presentation below to review the techniques for determining limits.

Squeeze Theorem

The Squeeze Theorem states that if f(x) < g(x) < h(x) when x is near a (except for possibly at a) and $LaTeX: \lim_{x\to a}f\left(x\right)=\lim_{x\to a}h\left(x\right)=L$ $\lim_{x\to a}f\left(x\right)=\lim_{x\to a}h\left(x\right)=L$ equation image indicator , then $LaTeX: \lim_{x\to a}g\left(x\right)=L$ $\lim_{x\to a}g\left(x\right)=L$ . The Squeeze Theorem is sometimes referred to as the Sandwich Theorem. View the presentation on an intuitive proof of the Squeeze Theorem.

Special Trigonometric Limits

Recall earlier in this module we investigated equation image indicator $LaTeX: \lim_{x\to 0}\frac{sin x}{x}$ $\lim_{x\to 0}\frac{sin x}{x}$ from graphical and numerical approaches. View the presentation below of a proof of $LaTeX: \lim_{x\to 0}\frac{sin x}{x}=1$ $\lim_{x\to 0}\frac{sin x}{x}=1$ .

View the presentation below on how the Squeeze Theorem is used to develop other special trigonometric limits.

View the presentation below illustrating how to find limits using special trigonometric limits.

Evaluating Limits Analytically Practice

Evaluating Limits Analytically: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of how to employ a variety of techniques to evaluate limits analytically.

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