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Evaluating the limit of a function algebraically

Evaluate lim x 3 ( 2 x + 5 ) .

lim x 3 ( 2 x + 5 ) = lim x 3 ( 2 x ) + lim x 3 ( 5 ) Sum of functions property                       = 2 lim x 3 ( x ) + lim x 3 ( 5 ) Constant times a function property                       = 2 ( 3 ) + 5   Evaluate                       = 11
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Evaluate the following limit: lim x 12 ( 2 x + 2 ) .

26

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Finding the limit of a polynomial

Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit    of a polynomial function as x approaches a is equivalent to simply evaluating the function for a .

Given a function containing a polynomial, find its limit.

  1. Use the properties of limits to break up the polynomial into individual terms.
  2. Find the limits of the individual terms.
  3. Add the limits together.
  4. Alternatively, evaluate the function for a .

Evaluating the limit of a function algebraically

Evaluate lim x 3 ( 5 x 2 ) .

lim x 3 ( 5 x 2 ) = 5 lim x 3 ( x 2 ) Constant times a function property                  = 5 ( 3 2 ) Function raised to an exponent property                  = 45
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Evaluate lim x 4 ( x 3 5 ) .

59

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Evaluating the limit of a polynomial algebraically

Evaluate lim x 5 ( 2 x 3 3 x + 1 ) .

lim x 5 ( 2 x 3 3 x + 1 ) = lim x 5 ( 2 x 3 ) lim x 5 ( 3 x ) + lim x 5 ( 1 ) Sum of functions                                 = 2 lim x 5 ( x 3 ) 3 lim x 5 ( x ) + lim x 5 ( 1 ) Constant times a function                                 = 2 ( 5 3 ) 3 ( 5 ) + 1 Function raised to an exponent                                 = 236 Evaluate
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Evaluate the following limit: lim x 1 ( x 4 4 x 3 + 5 ) .

10

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Finding the limit of a power or a root

When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit    of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.

Evaluating a limit of a power

Evaluate lim x 2 ( 3 x + 1 ) 5 .

We will take the limit of the function as x approaches 2 and raise the result to the 5 th power.

lim x 2 ( 3 x + 1 ) 5 = ( lim x 2 ( 3 x + 1 ) ) 5                        = ( 3 ( 2 ) + 1 ) 5                        = 7 5                        = 16,807
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Evaluate the following limit: lim x 4 ( 10 x + 36 ) 3 .

64

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If we can’t directly apply the properties of a limit, for example in lim x 2 ( x 2 + 6 x + 8 x 2 ) , can we still determine the limit of the function as x approaches a ?

Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.

Finding the limit of a quotient

Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.

Practice Key Terms 1

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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