Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→b (x − b)^50 − x + b / x − b

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Identify the form of the limit: As \( x \to b \), the expression \((x - b)^{50} - x + b\) becomes \(0 - b + b = 0\), and the denominator \(x - b\) also becomes 0, indicating an indeterminate form \(\frac{0}{0}\).

Apply L'Hôpital's Rule: Since the limit is in the indeterminate form \(\frac{0}{0}\), we can apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately.

Differentiate the numerator: The derivative of \((x - b)^{50} - x + b\) with respect to \(x\) is \(50(x - b)^{49} - 1\).

Differentiate the denominator: The derivative of \(x - b\) with respect to \(x\) is 1.

Evaluate the new limit: Substitute \(x = b\) into the expression \(\frac{50(x - b)^{49} - 1}{1}\) 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 how functions behave near points of interest, including points where they may not be defined. Evaluating limits is crucial 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 ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex cases.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. In the context of limits, polynomial functions are continuous everywhere, which means their limits can often be evaluated by direct substitution. Understanding the behavior of polynomials as they approach specific values is essential for solving limit problems.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.lim x→b (x − b)^50 − x + b / x − b