Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→2 (x² - 2x / (x² - 6x + 8)

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First, substitute x = 2 into the expression to check if the limit results in an indeterminate form. Calculate the numerator: x² - 2x = 2² - 2(2) = 0. Calculate the denominator: x² - 6x + 8 = 2² - 6(2) + 8 = 0. Since both the numerator and denominator are zero, the limit is in the indeterminate form 0/0, so l'Hôpital's Rule can be applied.

Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. Differentiate the numerator: d/dx(x² - 2x) = 2x - 2.

Differentiate the denominator: d/dx(x² - 6x + 8) = 2x - 6.

Now, substitute x = 2 into the new expression obtained after applying l'Hôpital's Rule: (2x - 2) / (2x - 6).

Evaluate the limit of the new expression as x approaches 2. Substitute x = 2 into the expression: (2(2) - 2) / (2(2) - 6). Simplify the expression to find the limit.

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be defined. Evaluating limits is essential for determining continuity, derivatives, and integrals.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This technique is often used in limit problems to simplify expressions, especially when evaluating limits at points where the function is undefined, allowing for easier computation.

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→2 (x² - 2x / (x² - 6x + 8)