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#Finding limits in calculus rules how to#
#Finding limits in calculus rules license#

#Finding limits in calculus rules license#

Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License In Example 2.19, we look at simplifying a complex fraction. Example 2.17 illustrates the factor-and-cancel technique Example 2.18 shows multiplying by a conjugate. The next examples demonstrate the use of this Problem-Solving Strategy.

If f ( x ) / g ( x ) f ( x ) / g ( x ) is a complex fraction, we begin by simplifying it.

If the numerator or denominator contains a difference involving a square root, we should try multiplying the numerator and denominator by the conjugate of the expression involving the square root.

If f ( x ) f ( x ) and g ( x ) g ( x ) are polynomials, we should factor each function and cancel out any common factors.

To do this, we may need to try one or more of the following steps:

We then need to find a function that is equal to h ( x ) = f ( x ) / g ( x ) h ( x ) = f ( x ) / g ( x ) for all x ≠ a x ≠ a over some interval containing a.

First, we need to make sure that our function has the appropriate form and cannot be evaluated immediately using the limit laws.

Problem-Solving Strategy: Calculating a Limit When f ( x ) / g ( x ) f ( x ) / g ( x ) has the Indeterminate Form 0/0

lim x → 2 x + lim x → 2 1 ( lim x → 2 x ) 3 + lim x → 2 4 Apply the power law.

lim x → 2 2 x 2 − 3 x + 1 x 3 + 4 = lim x → 2 ( 2 x 2 − 3 x + 1 ) lim x → 2 ( x 3 + 4 ) Apply the quotient law, making sure that.

lim x → 2 x + lim x → 2 1 lim x → 2 x 3 + lim x → 2 4 Apply the sum law and constant multiple law.

Lim x → 2 2 x 2 − 3 x + 1 x 3 + 4 = lim x → 2 ( 2 x 2 − 3 x + 1 ) lim x → 2 ( x 3 + 4 ) Apply the quotient law, making sure that. Again, we need to keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to be applied. To find this limit, we need to apply the limit laws several times.

We now practice applying these limit laws to evaluate a limit. Root law for limits: lim x → a f ( x ) n = lim x → a f ( x ) n = L n lim x → a f ( x ) n = lim x → a f ( x ) n = L n for all L if n is odd and for L ≥ 0 L ≥ 0 if n is even and f ( x ) ≥ 0 f ( x ) ≥ 0. Power law for limits: lim x → a ( f ( x ) ) n = ( lim x → a f ( x ) ) n = L n lim x → a ( f ( x ) ) n = ( lim x → a f ( x ) ) n = L n for every positive integer n. Quotient law for limits: lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = L M lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) = L M for M ≠ 0 M ≠ 0 Product law for limits: lim x → a ( f ( x ) lim x → a f ( x ) = c L lim x → a c f ( x ) = c.Sum law for limits: lim x → a ( f ( x ) + g ( x ) ) = lim x → a f ( x ) + lim x → a g ( x ) = L + M lim x → a ( f ( x ) + g ( x ) ) = lim x → a f ( x ) + lim x → a g ( x ) = L + Mĭifference law for limits: lim x → a ( f ( x ) − g ( x ) ) = lim x → a f ( x ) − lim x → a g ( x ) = L − M lim x → a ( f ( x ) − g ( x ) ) = lim x → a f ( x ) − lim x → a g ( x ) = L − MĬonstant multiple law for limits: lim x → a c f ( x ) = c Then, each of the following statements holds: Assume that L and M are real numbers such that lim x → a f ( x ) = L lim x → a f ( x ) = L and lim x → a g ( x ) = M. Let f ( x ) f ( x ) and g ( x ) g ( x ) be defined for all x ≠ a x ≠ a over some open interval containing a. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions. The first two limit laws were stated in Two Important Limits and we repeat them here. These two results, together with the limit laws, serve as a foundation for calculating many limits. We begin by restating two useful limit results from the previous section. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes.

#Finding limits in calculus rules how to#

In this section, we establish laws for calculating limits and learn how to apply these laws. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values.

2.3.6 Evaluate the limit of a function by using the squeeze theorem.

2.3.5 Evaluate the limit of a function by factoring or by using conjugates.

2.3.4 Use the limit laws to evaluate the limit of a polynomial or rational function.

2.3.3 Evaluate the limit of a function by factoring.

2.3.2 Use the limit laws to evaluate the limit of a function.

If the left-hand limit and a right-hand limit of a function both exist for a particular value and are the same, then the function is said to have a two-sided limit at that value.