Problem 37 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow b} \frac{(x-b)^{50}-x+b}{x-b}$$

Step-by-Step Solution

Answer

Question: Find the limit of the given rational function as \(x\) approaches \(b\): $$\lim _{x \rightarrow b} \frac{(x-b)^{50}-x+b}{x-b}$$ Answer: The limit of the given expression as \(x\) approaches \(b\) is -1.

Step 1: Rewrite the expression in the limit

Let's first rewrite the expression within the limit: $$\lim _{x \rightarrow b} \frac{(x-b)^{50}-x+b}{x-b}$$

Step 2: Factor out \((x - b)\) from the numerator

To do this, we can use substitution and polynomial factoring. Let \(y = x - b\). Then we can rewrite the expression as: $$\lim _{x \rightarrow b} \frac{y^{50}- (y + b) + b}{y}$$ Next, let's factor out \(y\) from the terms in the numerator: $$\lim _{x \rightarrow b} \frac{y(y^{49} - 1)}{y}$$

Step 3: Simplify the expression

Now, we can cancel out the \(y\) terms from the numerator and denominator. This results in: $$\lim _{x \rightarrow b} (y^{49} - 1)$$

Step 4: Substitute \(x - b\) back into the expression

We now substitute back \(x - b\) for \(y\): $$\lim _{x \rightarrow b} ((x-b)^{49} - 1)$$

Step 5: Evaluate the limit

Finally, we can evaluate the limit as \(x\) approaches \(b\): $$\lim _{x \rightarrow b} ((x-b)^{49} - 1) = ((b-b)^{49} - 1) = (0^{49} - 1) = -1$$ Therefore, the limit is: $$\lim _{x \rightarrow b} \frac{(x-b)^{50}-x+b}{x-b} = -1$$

Key Concepts

Factorization

Factorization is a mathematical technique where we express an equation or expression as a product of its factors. In calculus and algebra, factorization can simplify complex expressions, making it easier to solve equations or evaluate limits.

• By factoring out common elements within the numerator or denominator, it is easier to make cancellations that reduce the expression down to a simpler form.
• For instance, in the problem above, the expression \((x-b)^{50} - x + b\) involves large algebraic power which gets simplified by recognizing a factor of \((x-b)\).
• Performing these factorizations provides a clearer pathway to evaluate limits and solve for unknowns.

Factorization is especially handy when dealing with polynomial expressions, where identifying and removing repetitive terms leads to a much simpler expression. It also directly leads to cancelling out terms, as seen in our problem where the terms with \(y\) can be cancelled from the numerator and denominator.

Limit Evaluation

Limit evaluation in calculus involves determining the value that a function approaches as the input approaches a certain point. It is an essential concept in understanding the behavior of functions, especially when grappling with undefined expressions.

• The exercise requires evaluating a limit where direct substitution initially leads to an indeterminate form, e.g., something resembling \( \frac{0}{0} \).
• To navigate this, manipulations such as factoring, simplification, or applying L'Hopital's rule are employed to resolve these forms.
• In the provided solution, factorization and cancellation allowed a direct substitution after simplification.

The essence of limit evaluation is in transforming the expression to a state where direct substitution can be applied, revealing the behavior near the point of interest. Here, once the terms were simplified, the limit could be directly evaluated as -1, illustrating the importance of these techniques.

Polynomial Expressions

Polynomial expressions are mathematical expressions consisting of variables and coefficients, structured as sums of terms with non-negative integer exponents. These expressions often appear in calculus problems, including those dealing with limits.

• Understanding polynomial behavior, such as recognizing patterns in their expansion, is key to simplifying complex expressions.
• In the given problem, the polynomial form is expressed as \((x-b)^{50}\), indicating the need for simplification to evaluate the limit effectively.
• By substituting \(y = x - b\), the polynomial takes a different form, aiding further simplification.

Polynomials can be intimidating due to their potentially high degree, but techniques like factoring, regrouping, or using substitutions similar to those used in the exercise, significantly simplify their evaluation. Mastering these expressions is crucial for solving higher-level calculus problems efficiently.