Question

Determine the following limits at infinity. $$\lim _{x \rightarrow \infty}\left(3+\frac{10}{x^{2}}\right)$$

Short answer

Answer: The limit of the expression $$3+\frac{10}{x^2}$$ as x approaches infinity is 3.

Step-by-step solution

Identify the dominant term

As x approaches infinity, the expression is dominated by the constant 3, since the fraction $$\frac{10}{x^2}$$ will approach zero.

Evaluate the limit of the dominant term

The constant 3 does not depend on x, so its limit as x approaches infinity is simply 3.

Evaluate the limit of the fraction

As x approaches infinity, the fraction $$\frac{10}{x^2}$$ approaches zero, because the denominator grows without bounds and the numerator remains constant.

Combine the limits

The overall limit of the expression can be found by adding the limits of its components: $$\lim _{x \rightarrow \infty}\left(3+\frac{10}{x^{2}}\right) = \lim _{x \rightarrow \infty} 3 + \lim _{x \rightarrow \infty} \left(\frac{10}{x^2}\right)$$

Compute the final limit

Since the limit of the dominant term is 3, and the limit of the fraction is 0, the overall limit is: $$\lim _{x \rightarrow \infty}\left(3+\frac{10}{x^{2}}\right) = 3 + 0 = 3$$

Key concepts

Dominant Term in Limits

When assessing the behavior of functions as they approach infinity, the dominant term is a crucial concept to identify. The dominant term is essentially the part of the function that grows the fastest and overshadows the impact of all other terms as the variable approaches infinity. For instance, in the exercise \( \lim _{x \rightarrow \infty}(3 + \frac{10}{x^{2}}) \), the term '3' is dominant compared to \( \frac{10}{x^{2}} \) because '3' remains constant regardless of the value of x, while \( \frac{10}{x^{2}} \) diminishes to zero.

Understanding which term dominates helps simplify the problem. As x becomes larger, less significant terms can be dismissed for the purpose of finding the limit. The larger the value of x, the smaller the impact of the non-dominant terms, ultimately leading to them having no effect on the outcome of the limit as x tends towards infinity.

Limit Properties

Linearity of Limits

The properties of limits are essential tools that allow us to break down complex expressions into simpler parts. One such property is the linearity of limits, which states that the limit of a sum is equal to the sum of the limits, assuming that the limits exist. In mathematical terms, \( \lim_{x \to \infty} (f(x) + g(x)) = \lim_{x \to \infty} f(x) + \lim_{x \to \infty} g(x) \). This property was used in step 4 of the solution, where the limit of the entire expression was divided into the limits of its individual components. Similarly, multiplication by a constant can be factored out of a limit and evaluated separately.

This linearity simplifies the evaluation process, allowing us to find the limits of more complicated functions by separating them into parts we can manage more easily. Moreover, this structural approach to evaluating limits ensures accurate results by addressing each part of an expression independently.

Limit Evaluation

The Evaluation of limits is the final step which provides the exact numerical value that a function approaches as the variable approaches a certain point or infinity. There are certain strategies employed in limit evaluation, such as identifying the dominant term, using limit properties, and applying known limits of basic functions.

In the given problem, we directly evaluate the limit of the constant '3', which remains unchanged, as constants have the same value regardless of the input. On the other hand, for \( \frac{10}{x^{2}} \), we recognize that as x approaches infinity, this term approaches zero since the numerator stays constant and the denominator grows increasingly larger. After finding these individual limits, we employ summing, which is justified by the limit properties, to give us the final evaluated limit of the function as 3. This demonstrates the harmonious connection between identifying the dominant term, using limit properties, and the actual evaluation of limits.