Math Problem Statement

Question content area top Part 1 Find the following limit or state that it does not exist. ModifyingBelow lim With x right arrow 2 left parenthesis 2 x cubed minus 4 x squared plus 2 x plus 5 right parenthesis

Solution

We need to evaluate the limit:

limx2(2x34x2+2x+5)\lim_{x \to 2} \left( 2x^3 - 4x^2 + 2x + 5 \right)

Step 1: Direct Substitution

Since this is a polynomial function, it is continuous for all real values of xx. This means we can directly substitute x=2x = 2:

2(2)34(2)2+2(2)+52(2)^3 - 4(2)^2 + 2(2) + 5

Step 2: Compute Each Term

2(8)4(4)+2(2)+52(8) - 4(4) + 2(2) + 5

1616+4+516 - 16 + 4 + 5

99

Conclusion

Thus, the limit is:

limx2(2x34x2+2x+5)=9\lim_{x \to 2} \left( 2x^3 - 4x^2 + 2x + 5 \right) = 9

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Related Questions:

  1. What is the general rule for finding the limit of a polynomial function?
  2. How does continuity help in evaluating limits?
  3. What happens when direct substitution leads to an indeterminate form?
  4. How do you evaluate limits of rational functions?
  5. What techniques can be used if direct substitution fails?

Tip: Always check for direct substitution first when evaluating limits, especially for polynomial functions!

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Math Problem Analysis

Mathematical Concepts

Limits
Polynomial Functions
Continuity
Substitution Method

Formulas

lim_{x→a} f(x) = f(a) for continuous functions

Theorems

Continuity Theorem
Direct Substitution Theorem

Suitable Grade Level

Grades 10-12