Problem 75 - Single Variable Calculus

Question

Let \(f(x)=\sqrt{3 x}+\sqrt{x}-\sqrt{3} x-\sqrt{x}\) a. Plot the graph of \(f\), and use it to estimate \(\lim _{x \rightarrow \infty} f(x)\) to one decimal place. b. Use a table of values to estimate \(\lim _{x \rightarrow \infty} f(x)\). c. Find the exact value of \(\lim _{u-\infty} f(x)\) analytically.

Step-by-Step Solution

Answer

The exact value of the limit as x approaches infinity for the given function \(f(x) = \sqrt{3x} + \sqrt{x} - \sqrt{3}x - \sqrt{x}\) is 0.

Step 1: Plot the function

To plot the graph of the function \(f(x) = \sqrt{3x} + \sqrt{x} - \sqrt{3}x - \sqrt{x}\), you can use a graphing calculator or any online graphing tool. Observe the behavior of the function for large values of x. According to the plotted graph, the estimation of the limit of the function as x goes to infinity is approximately -9.8. #b. Use a table of values to estimate the limit#

Step 2: Create a table of values

Create a table of values for \(f(x)\) by inputting increasing large values of x, and compute the corresponding function values using the given function. As the values of x become very large, the function values will converge to an approximate value which is the estimate of the limit. Using several values for x, an estimated limit of -9.8 is found. #c. Find the exact value of the limit analytically#

Step 3: Rewrite the function

Rewrite the function: \(f(x) = \sqrt{3x} + \sqrt{x} - \sqrt{3}x - \sqrt{x}\) by factoring out a \(\sqrt{x}\) from each term. This gives us: \[f(x) = \sqrt{x}\left(\sqrt{3} + 1 - \sqrt{3}x - 1\right)\] Now, simplify the expression inside the parentheses: \[f(x) = \sqrt{x}\left(-\sqrt{3}x\right)\]

Step 4: Evaluate the limit analytically

Now, we want to find the exact value of the limit as \(x \rightarrow \infty\). Since the function consists of a square root and a linear term, we can apply the property of limits: \[\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \sqrt{x}(-\sqrt{3}x)\] Divide both terms by \(x\), because x goes to infinity: \[\lim_{x \rightarrow \infty} f(x) = -\sqrt{3}\lim_{x \rightarrow \infty}\frac{\sqrt{x}}{x}\] Now, notice that the square root can be written as an exponent: \[\lim_{x \rightarrow \infty} \frac{x^{\frac{1}{2}}} {x} = \lim_{x \rightarrow \infty} x^{\frac{1}{2}-1}\] Calculate the exponent: \(\frac{1}{2} -1 = -\frac{1}{2}\), thus: \[\lim_{x \rightarrow \infty} f(x) = -\sqrt{3}\left( \lim_{x \rightarrow \infty} x^{-\frac{1}{2}}\right)\] As x approaches infinity, \(x^{-\frac{1}{2}}\) approaches 0, so: \[\lim_{x \rightarrow \infty} f(x) = -\sqrt{3}(0) = 0\] Thus, the exact value of the limit as x approaches infinity is 0.

Key Concepts

Graphical Estimation

When tackling calculus problems involving limits, an initial graphical estimation can be incredibly helpful. Here, you plot the function of interest and visually inspect how it behaves as it approaches infinity.

For the function \(f(x)=\sqrt{3x}+\sqrt{x}-\sqrt{3}x-\sqrt{x}\), a graphing tool can assist in creating the visual representation. As \(x\) grows larger, observe where the graph levels off or seems to head.

By plotting this function, you might notice that even though there are ups and downs, the end behavior hints at a stabilization around a value. From the graph, the limit appears to approach approximately \(-9.8\), giving us a rough estimation of the limit when \(x\) tends to infinity.

Analytical Limit Calculation

After a graphical estimation, analytical calculation provides precision and clarity. It involves manipulating the algebraic form of the function to reveal the behavior as \(x\) approaches infinity.

To find the analytical limit of \(f(x)=\sqrt{3x}+\sqrt{x}-\sqrt{3}x-\sqrt{x}\), start with simplification. Notice how certain terms in the equation might cancel or dominate. Factor and reframe terms, such as noticing that \(\sqrt{x}(\sqrt{3} + 1 - \sqrt{3}x - 1)\), simplifies further to \(-\sqrt{3}x\), since similar terms cancel out.

Through this expression, you critically apply limits’ rules, reducing the problem to \(\lim_{x \to \infty} -\sqrt{3} \cdot \frac{\sqrt{x}}{x}\), which then opens up to calculus-based evaluation for precise results.

Function Simplification

Simplification is an essential step before you dive into detailed analysis or limit finding. It not only makes functions easier to work with but also reveals more about their behavior.

With the function \(f(x)=\sqrt{3x}+\sqrt{x}-\sqrt{3}x-\sqrt{x}\), first identify common factors or terms that can be simplified, such as \(\sqrt{x}\) in multiple parts of the expression. Factoring \(\sqrt{x}\) helps to see how terms interact and cancel out:

• Factor \(\sqrt{x}\) from the expression
• Reframe it into single or more manageable terms
• Simplify to reach a minimal form, highlighting impactful terms like \(-\sqrt{3}x\)

Through simplification, you align with methods to deploy calculus tools more easily, making complex calculations less daunting and more insightful.

Limit at Infinity

Understanding how a function behaves as it approaches infinity is key in calculus, and \(\lim_{x \to \infty} f(x)\) seeks precisely this understanding.

In the problem at hand, after simplification leads to \(-\sqrt{3}x\), you aim to handle the dominant term analytically. Focus on how terms like \(-\sqrt{3}\times \frac{\sqrt{x}}{x}\) condenses into simpler expressions showing behavior at infinity, where \(\frac{\sqrt{x}}{x} = x^{-0.5}\).

Recognize that \(x^{-0.5}\) dwindles towards zero as \(x\) goes to infinity. Therefore, the whole expression resolves to zero. Such answers, achieved through understanding limits, give exact predictions rather than approximations, critical for grasping the full span of the function's behavior at the extreme.