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Zorluk: KolayLimits and Continuity of Functions

Evaluate the limit lim⁡x→3x2−5x+6x−3\lim_{x \to 3} \frac{x^2 - 5x + 6}{x - 3}. What is the numerical value of this limit?

Cevap: 1

Cevap

The numerical value of the limit is 1.
Substituting x = 3 into the original expression produces the indeterminate form 0/0. Factoring the numerator gives (x - 3)(x - 2). Canceling the common factor (x - 3) yields x - 2. Substituting x = 3 into x - 2 gives 3 - 2 = 1.

Adım Adım Çözüm

1
Check the expression using direct substitution at x = 3.
The substitution yields \frac{3^2 - 5(3) + 6}{3 - 3} = \frac{0}{0}, an indeterminate form.
Direct substitution results in 0/0, indicating that algebraic factorization is needed.
2
Factorize the quadratic polynomial in the numerator.
x^2 - 5x + 6 = (x - 3)(x - 2)
Finding factors of +6 that sum to -5 allows cancellation of the denominator term.
3
Cancel the factor (x - 3) and evaluate the remaining linear expression at x = 3.
\lim_{x \to 3} (x - 2) = 3 - 2 = 1
The factor causing zero in the denominator is removed, leaving a continuous function.

Anahtar Kavram

Evaluating indeterminate limits (0/0) by algebraic factorization
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Konular

Limits and Continuity of Functions

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