Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)

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Question: Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)

Options:

Correct Answer: Undefined

Exam Year: 2021

Solution:

The expression is undefined at x=2. The limit does not exist as the function approaches infinity.

Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)

Find the limit: lim (x -> 2) (x^2 - 3x + 2)/(x - 2) (2021)

Step 1: Identify the limit we need to find: lim (x -> 2) (x^2 - 3x + 2)/(x - 2).
Step 2: Substitute x = 2 into the expression. This gives us (2^2 - 3*2 + 2)/(2 - 2) = (4 - 6 + 2)/(0) = 0/0, which is undefined.
Step 3: Since we have an undefined form (0/0), we need to simplify the expression.
Step 4: Factor the numerator: x^2 - 3x + 2 can be factored as (x - 1)(x - 2).
Step 5: Rewrite the limit: lim (x -> 2) ((x - 1)(x - 2))/(x - 2).
Step 6: Cancel the (x - 2) terms in the numerator and denominator, but note that this is valid only for x ≠ 2.
Step 7: Now the limit simplifies to lim (x -> 2) (x - 1).
Step 8: Substitute x = 2 into the simplified expression: 2 - 1 = 1.
Step 9: Therefore, the limit is 1.

Limits – Understanding how a function behaves as it approaches a certain point, particularly when the function is undefined at that point.
Factoring – The ability to factor polynomials to simplify expressions and find limits.
Indeterminate Forms – Recognizing when a limit results in an indeterminate form, such as 0/0, and applying techniques to resolve it.