Math Problem Statement

Find the indicated limit.

lim t→2 (6t2 − 3t + 1)

Solution

To find the limit of the given function as t2t \to 2:

limt2(6t23t+1)\lim_{t \to 2} (6t^2 - 3t + 1)

We can substitute t=2t = 2 directly into the expression, since the function is a polynomial and continuous everywhere.

6(2)23(2)+1=6(4)6+1=246+1=196(2)^2 - 3(2) + 1 = 6(4) - 6 + 1 = 24 - 6 + 1 = 19

Thus, the limit is:

limt2(6t23t+1)=19\lim_{t \to 2} (6t^2 - 3t + 1) = 19

Would you like a more detailed explanation or have any questions?

Here are five related questions for further practice:

  1. What is limt3(5t22t+4)\lim_{t \to 3} (5t^2 - 2t + 4)?
  2. How would you evaluate limx1(x3+4x2x+6)\lim_{x \to -1} (x^3 + 4x^2 - x + 6)?
  3. What is the limit of limx0(2x23x+1)\lim_{x \to 0} (2x^2 - 3x + 1)?
  4. Can you compute limt5(3t2+7t2)\lim_{t \to 5} (3t^2 + 7t - 2)?
  5. How do you find limx2(4x32x+3)\lim_{x \to 2} (4x^3 - 2x + 3)?

Tip: Polynomials are continuous everywhere, so direct substitution is a valid method for finding limits involving polynomials.

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Math Problem Analysis

Mathematical Concepts

Limits
Polynomials
Continuity

Formulas

Direct substitution for limits of polynomials: lim t→c f(t) = f(c)

Theorems

Polynomial Continuity Theorem

Suitable Grade Level

Grades 10-12