Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)

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Question: Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)

Options:

Correct Answer: Undefined

Solution:

Factoring gives (x(x - 2))/(x - 2), canceling gives lim (x -> 2) x = 2.

Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)

Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)

Step 1: Identify the limit you need to calculate: lim (x -> 2) (x^2 - 2x)/(x - 2).
Step 2: Notice that if you directly substitute x = 2, both the numerator and denominator become 0, which is an indeterminate form.
Step 3: Factor the numerator (x^2 - 2x). It can be factored as x(x - 2).
Step 4: Rewrite the limit using the factored form: lim (x -> 2) (x(x - 2))/(x - 2).
Step 5: Cancel the (x - 2) terms in the numerator and denominator, as long as x is not equal to 2.
Step 6: Now the limit simplifies to lim (x -> 2) x.
Step 7: Substitute x = 2 into the simplified expression: 2.
Step 8: Conclude that the limit is 2.

Limits – Understanding how to evaluate limits, especially when direct substitution leads to an indeterminate form.
Factoring – The ability to factor expressions to simplify limits and resolve indeterminate forms.
Cancellation – Recognizing when and how to cancel common factors in a limit expression.