Problem 91 - Calculus Early Transcendentals

Question

Given the polynomial $$ p(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0}, $$ prove that \(\lim _{x \rightarrow a} p(x)=p(a)\) for any value of \(a\).

Step-by-Step Solution

Answer

Based on the step by step solution given above: Question: Prove that for any polynomial function \(p(x)\) and any value of \(a\), the limit of the function as \(x\) approaches \(a\) is equal to the value of the function at that point, i.e., \(\lim _{x \rightarrow a} p(x)=p(a)\). Answer: To prove this, we first analyze the limit of each term in the polynomial, which are of the forms \(b_{i}x^{i}\). Then, using the limit properties like the constant multiple rule and the power rule, we simplify the expression to get \(b_{i} a^{i}\). Next, we find the limit of the entire polynomial by adding the limits of individual terms, and we substitute the simplified limit of each term back into our current expression. Finally, we see that the resulting expression is equal to the value of the polynomial function at the point \(a\), thus proving \(\lim _{x \rightarrow a} p(x) = p(a)\).

Step 1: Analyzing the limit of each term in the polynomial

We'll look at each term in the polynomial individually and evaluate its limit as \(x\) approaches \(a\). The terms in polynomial are of the forms \(b_{i}x^{i}\), where \(i\) varies from \(0\) to \(n\), and \(b_i\) are constants. Let's consider the limit of a general term, \(b_{i}x^{i}\), as \(x\) approaches \(a\): $$ \lim _{x \rightarrow a} b_{i} x^{i} $$

Step 2: Using limit properties

We can use the limit properties to manipulate the expression and make it simpler to evaluate. The properties we will use include: 1. Constant multiple rule: \(\lim _{x \rightarrow a} (k f(x)) = k (\lim _{x \rightarrow a} f(x))\), where \(k\) is a constant. 2. Power rule: \(\lim _{x \rightarrow a} x^{n} = a^{n}\), where \(n\) is a positive integer. Applying the constant multiple rule to the expression, we get: $$ \lim _{x \rightarrow a} b_{i} x^{i} = b_{i} (\lim _{x \rightarrow a} x^{i}) $$ Now, applying the power rule to the expression, we get: $$ b_{i} (\lim _{x \rightarrow a} x^{i}) = b_{i} a^{i} $$

Step 3: Finding the limit of the entire polynomial

Now that we've determined the limit of each term, we can find the limit of the entire polynomial. The limit of a sum (or difference) is the sum (or difference) of the individual limits. So, we can write: $$ \lim _{x \rightarrow a} p(x) = \lim _{x \rightarrow a} \left(b_{n}x^{n}+b_{n-1}x^{n-1}+\cdots+b_{1}x+b_{0}\right) $$ Using the sum/difference limit rules, this becomes: $$ \lim _{x \rightarrow a} p(x) = \lim _{x \rightarrow a} b_{n} x^{n} + \lim _{x \rightarrow a} b_{n-1} x^{n-1} + \cdots + \lim _{x \rightarrow a} b_{1} x + \lim _{x \rightarrow a} b_{0} $$

Step 4: Combining the limits

Using the results from the previous steps, we can now substitute the limit of each term back into our current expression: $$ \lim _{x \rightarrow a} p(x) = b_{n} a^{n} + b_{n-1} a^{n-1} + \cdots + b_{1} a + b_{0} $$

Step 5: Concluding the proof

We can now see that the expression on the right side is equal to the value of the polynomial function \(p(x)\) at the point \(a\): $$ \lim _{x \rightarrow a} p(x) = p(a) $$ This completes the proof that \(\lim _{x \rightarrow a} p(x)=p(a)\) for any value of \(a\).

Key Concepts

Limit Properties

Understanding limit properties is crucial when dealing with polynomials and calculus. Essentially, limits help us determine the value that a function approaches as we get closer to a certain point. Among these properties, the two often utilized in solving polynomial limits are the sum limit rule and the constant multiple rule. These properties allow us to break down complex expressions into more manageable parts, simplifying the computation of limits.

• Sum limit rule: The limit of a sum is the sum of the limits. This property is particularly useful in polynomials with multiple terms. It ensures that instead of assessing the limit of an entire polynomial at once, each term can be addressed individually.
• Constant multiple rule: This property states that the limit of a constant multiplied by a function is the constant multiplied by the limit of the function. When applied to polynomial terms, it makes evaluating the limits of terms straightforward, even when constants are present.

By using these properties, you can efficiently simplify and solve for the limits in polynomial functions.

Constant Multiple Rule

The constant multiple rule simplifies the process of finding limits where a constant is involved. When you encounter expressions like \( b_i x^i \), notice that the number \( b_i \) remains unaffected by the variable and can be factored out of the limit process.

• Imagine we have an expression \( \lim_{x ightarrow a} kf(x) \) and \( k \) is a constant.
• The rule allows us to reorganize this as \( k \cdot \lim_{x ightarrow a} f(x) \).
• For a polynomial term such as \( b_i x^i \), this turns into \( b_i \cdot \lim_{x \rightarrow a} x^i \).

This makes working with polynomial limits much simpler, as every constant can be easily pulled out from the calculation, leaving only the variable expressions to consider. Understanding this rule is key to mastering polynomial limits and makes the computation more intuitive.

Power Rule

The power rule is fundamental when determining the limit of polynomials. This rule states that for any expression \( x^n \), as \( x \) approaches \( a \), the limit becomes \( a^n \). Consider the expression \( \lim_{x \rightarrow a} x^i \).

• The power rule helps us directly substitute \( a \) for \( x \) in the expression, yielding \( a^i \) as the result.
• This convenience means that for any polynomial term \( x^n \), evaluating the term's limit is straightforward and doesn't require complicated calculations.

These principles are powerful because they transform what could be a complex task into a relatively simple substitution exercise. When combined with other limit properties, the power rule is an invaluable tool in calculus.

Sum Limit Rule

The sum limit rule is a straightforward yet powerful concept that greatly facilitates the calculation of limits for polynomials and other functions. According to this rule, the limit of a sum of functions equals the sum of their individual limits. Let's see how this works with polynomials:

• For a polynomial function \( p(x) = b_n x^n + b_{n-1} x^{n-1} + \cdots + b_1 x + b_0 \), we can split the limit into separate terms using this rule.
• For each term, the limit operation is distributed, giving us \( \lim_{x \rightarrow a} b_n x^n + \lim_{x \rightarrow a} b_{n-1} x^{n-1} + \cdots + \lim_{x \rightarrow a} b_1 x + \lim_{x \rightarrow a} b_0 \).
• This separation allows for simpler calculation and comprehensive assessment of each term individually.

This property is instrumental in breaking down polynomials to more basic calculations, allowing an easy approach to otherwise complex problems. It ensures that instead of handling an entire polynomial, we can focus on evaluating one term at a time.