Question

\(\operatorname{Lt}_{x \rightarrow \infty} \frac{(x+2)^{10}+(x+4)^{10}+\ldots . .+(x+20)^{10}}{x^{10}+1}=\) (1) 12 (2) 20 (3) 40 (4) 10

Short answer

\(\operatorname{Lt}_{x \rightarrow \infty} \frac{(x+2)^{10}+(x+4)^{10}+...+(x+20)^{10}}{x^{10} + 1}\) (1) 0 (2) 1 (3) 5 (4) 10 (5) Does not exist Answer: (4) 10

Step-by-step solution

Find the degree of the numerator and denominator

The degree of each term in the numerator is 10 since every term is of the form \((x+n)^{10}\) where n is an integer value from 2 to 20. Since they are all summed together, the highest-degree term in the numerator will be \(x^{10}\). On the denominator side, we have \(x^{10} + 1\), and the highest degree is also 10.

Divide numerator and denominator by \(x^{10}\)

Divide each term in the numerator by \(x^{10}\) and also divide the denominator by \(x^{10}\). The given expression becomes: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{\frac{(x+2)^{10}}{x^{10}}+\frac{(x+4)^{10}}{x^{10}}+\ldots . .+\frac{(x+20)^{10}}{x^{10}}}{\frac{x^{10}+1}{x^{10}}} \)

Simplify the expression

Now let's break the given expression into multiple limits and use limit properties: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{\frac{(x+2)^{10}}{x^{10}}+\frac{(x+4)^{10}}{x^{10}}+\ldots . .+\frac{(x+20)^{10}}{x^{10}}}{\frac{x^{10}+1}{x^{10}}} = \operatorname{Lt}_{x \rightarrow \infty} \frac{\frac{(x+2)^{10}}{x^{10}}} + \operatorname{Lt}_{x \rightarrow \infty} \frac{\frac{(x+4)^{10}}{x^{10}}} + ... + \operatorname{Lt}_{x \rightarrow \infty} \frac{\frac{(x+20)^{10}}{x^{10}}}{\operatorname{Lt}_{x \rightarrow \infty} \frac{x^{10}+1}{x^{10}}}\) Now we can simplify the limits: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{(1 + \frac{2}{x})^{10} + (1 + \frac{4}{x})^{10} + \ldots . .+(1 + \frac{20}{x})^{10}}{1 + \frac{1}{x^{10}}}\)

Compute the limit

As x approaches infinity, \(\frac{2}{x}\), \(\frac{4}{x}\), ..., \(\frac{20}{x}\) and \(\frac{1}{x^{10}}\) all approach 0. Therefore, the limit becomes: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{1^{10} + 1^{10} + \ldots . .+ 1^{10}}{1 + 0}\) There are 10 terms in the numerator (as the sequence starts from \((x+2)^{10}\) and ends at \((x+20)^{10}\), with a gap of 2 between each term), and each term evaluates to 1: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{1 + 1 + \ldots . .+ 1}{1} = \frac{10}{1} = 10\) So the limit is 10. The correct answer is (4) 10.

Key concepts

Degree of polynomials

Understanding the degree of polynomials is crucial when solving limit problems in calculus that involve them. The degree of a polynomial is the highest power of the variable in the expression. For example, consider the polynomial \((x+2)^{10}\). The degree of this term is 10 because the highest power of \(x\) is 10. In the given exercise, every term in the numerator is of the form \((x+n)^{10}\), where \(n\) is an integer from 2 to 20. Thus, every term has the same degree, 10.
The degree of a polynomial helps determine its behavior for very large values of \(x\) (approaching infinity). Specifically, when comparing the degrees of the numerator and the denominator, we can use this information to simplify expressions. When the degrees are equal, like in our problem (both the numerator and the denominator have a degree of 10), the limit often results in the ratio of the leading coefficients. As we see, understanding and identifying degrees is a helpful step when dealing with limits involving polynomials.

Simplifying expressions

When dealing with complex algebraic expressions, particularly those that appear as fractions, it’s important to simplify them for easier evaluation. In the given limit problem, we have a complicated fraction and our goal is to break it down to simpler components.
To simplify, we divide both the numerator and the denominator by the highest power of \(x\) found in the denominator, which in this scenario is \(x^{10}\). This step is useful because it helps eliminate the highest power terms and reduces other terms to simpler coefficients and expressions such as \(\frac{n}{x}\) which tend to zero as \(x\) approaches infinity.
Once the expression is simplified, we apply properties of limits to handle each term separately. By considering each part of the fraction after division, solving for limits becomes more straightforward. For example, terms that appear as \(\frac{n^k}{x^k}\) go to 0 as \(x\) grows larger, allowing us to disregard them from the final result.

Properties of limits

Calculus provides helpful properties of limits that make evaluating them more manageable, especially when encountering more complex expressions or polynomials. Some important properties include:

• **Limit of a Sum**: The limit of a sum of functions is equal to the sum of their limits.
• **Limit of a Quotient**: The limit of a fraction is the limit of the numerator divided by the limit of the denominator, provided the limit of the denominator is not zero.
• **Limits at Infinity**: As \(x\) approaches infinity, terms that have a form \(\frac{n}{x^k}\) will tend towards 0. This is vital when simplifying expressions by lopping off negligible terms.

These properties can be applied when solving the given exercise. After rewriting the expression in a simplified form, the limit properties allow for the evaluated sum in the numerator to be straightforwardly calculated by limiting each component separately. Once terms involving \(x\) vanish or reduce to simpler limits, the result is a clear numerical outcome. For this problem, the properties make it evident that each individual term's limit equals one, leading to a cumulative sum of 10, the value of the given expression's limit.