Question

\(\operatorname{Lt}_{x \rightarrow \infty} \frac{5 x^{2}}{\sqrt{25 x^{4}+13 x^{3}+14 x^{2}+17 x+6}}\) is (1) 1 (2) \(1 / 5\) (3) \(5 / 14\) (4) None of these

Short answer

Question: Find the limit as x approaches infinity of the function \(\frac{5x^2}{\sqrt{25x^4 + 13x^3 + 14x^2 + 17x + 6}}\). Answer: 1

Step-by-step solution

Divide numerator and denominator by the highest power of x present in the denominator.

We have the highest power of x to be \(x^4\) (inside the square root) in the denominator. Therefore, we will divide both the numerator and the denominator by \(x^4\). The given function becomes: \(f(x) = \frac{5x^2}{\sqrt{(x^4)(25 + \frac{13}{x} + \frac{14}{x^2} + \frac{17}{x^3} + \frac{6}{x^4})}}\) Now, let's simplify it.

Simplify the rational function.

After dividing both the numerator and the denominator by \(x^4\), the function can be simplified as: \(f(x) = \frac{5}{\sqrt{25 + \frac{13}{x} + \frac{14}{x^2} + \frac{17}{x^3} + \frac{6}{x^4}}}\)

Find the limit as x approaches infinity.

Now, we will evaluate the limit as x approaches infinity: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{5}{\sqrt{25 + \frac{13}{x} + \frac{14}{x^2} + \frac{17}{x^3} + \frac{6}{x^4}}}\) As x approaches infinity, the terms \(\frac{13}{x}\), \(\frac{14}{x^2}\), \(\frac{17}{x^3}\), and \(\frac{6}{x^4}\) in the denominator will approach 0. Hence, we have: \(\operatorname{Lt}_{x \rightarrow \infty} \frac{5}{\sqrt{25 + 0 + 0 + 0 + 0}} = \frac{5}{\sqrt{25}} = \frac{5}{5} = 1\) So, the limit as x approaches infinity is 1, which corresponds to option (1).

Key concepts

Limits and Continuity

Understanding limits is essential in calculus as they help in analyzing the behavior of functions as inputs approach a certain value. Specifically, when we talk about the limit at infinity, we are interested in what value the function approaches as the input grows without bound. The problem in question concerns the limit of a rational function as x approaches infinity, which is written as \[\begin{equation}\operatorname{Lt}_{x \rightarrow \infty} \end{equation}\]
When calculating such limits, a key principle is that if the degrees of the polynomial in the numerator and denominator are the same, the limit at infinity is the ratio of the leading coefficients.

Continuity, on the other hand, implies that a function is unbroken or uninterrupted. In terms of limits, a function f is continuous at a point a if the following are true:

• The function is defined at a.
• The limit of f as x approaches a exists.
• The limit as x approaches a is equal to f(a).

Rational Function Simplification

In calculus, rational function simplification is a valuable technique for finding limits. A rational function is a ratio of two polynomials. When dealing with limits at infinity, the degrees of the polynomials matter a great deal. In the given exercise, the simplification happens by dividing both the numerator and the denominator by the same power of x, specifically the highest power in the denominator.

This process helps us identify the dominant term — which is the term that has the greatest effect on the value of the function as x increases. By dividing both the numerator and the denominator by the highest power of x, we are able to cancel out the lesser terms that become insignificant as em>x grows large, leading to a more manageable expression that can be directly evaluated to find the limit at infinity.

Asymptotic Behavior

The concept of asymptotic behavior refers to the way a function behaves as it moves towards infinity or towards a point. Specifically, it describes the direction or trend that a function's value heads towards, as it gets closer to a line called an asymptote without ever touching it. Asymptotic behavior is a critical concept when dealing with limits at infinity because it explains what value a function approaches when the input gets larger and larger.

In the exercise provided, after simplifying the function, the remaining terms with a denominator containing x become negligible as x approaches infinity. Therefore, the function's value approaches the value of the simplified constant terms only. Here, since the remaining terms resolve to a constant, the horizontal line y = 1 is the horizontal asymptote indicating that the limit of the function as x approaches infinity is 1.