At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Ask your questions and receive precise answers from experienced professionals across different disciplines. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To find the limit \(\lim_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}}\), let's analyze the expression step by step.
### Step 1: Factor and Simplify
Consider the expression:
[tex]\[ \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}} \][/tex]
First, let's rewrite \(\sqrt[3]{x}\) as \(x^{1/3}\) and \(\sqrt[5]{x}\) as \(x^{1/5}\). The expression becomes:
[tex]\[ \frac{x^{1/3} - x^{1/5}}{x^{1/3} + x^{1/5}} \][/tex]
### Step 2: Factor Out the Dominant Term
The dominant term in both the numerator and denominator as \(x \to \infty\) is \(x^{1/3}\). Factor \(x^{1/3}\) out from both the numerator and the denominator:
[tex]\[ \frac{x^{1/3} (1 - x^{(1/5 - 1/3)})}{x^{1/3} (1 + x^{(1/5 - 1/3)})} \][/tex]
Notice that \(1/5\) is less than \(1/3\), so \(1/5 - 1/3\) is negative. Let’s simplify the exponent:
[tex]\[ \frac{x^{1/3} (1 - x^{-2/15})}{x^{1/3} (1 + x^{-2/15})} \][/tex]
### Step 3: Cancelling Out x^{1/3}
Now, cancel the common factor of \(x^{1/3}\) from both the numerator and the denominator:
[tex]\[ \frac{1 - x^{-2/15}}{1 + x^{-2/15}} \][/tex]
### Step 4: Evaluate the Limit
As \(x \to \infty\), \(x^{-2/15} \to 0\). Thus, the expression simplifies to:
[tex]\[ \frac{1 - 0}{1 + 0} = \frac{1}{1} = 1 \][/tex]
### Final Answer
Therefore, the limit is:
[tex]\[ \lim_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}} = 1 \][/tex]
### Step 1: Factor and Simplify
Consider the expression:
[tex]\[ \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}} \][/tex]
First, let's rewrite \(\sqrt[3]{x}\) as \(x^{1/3}\) and \(\sqrt[5]{x}\) as \(x^{1/5}\). The expression becomes:
[tex]\[ \frac{x^{1/3} - x^{1/5}}{x^{1/3} + x^{1/5}} \][/tex]
### Step 2: Factor Out the Dominant Term
The dominant term in both the numerator and denominator as \(x \to \infty\) is \(x^{1/3}\). Factor \(x^{1/3}\) out from both the numerator and the denominator:
[tex]\[ \frac{x^{1/3} (1 - x^{(1/5 - 1/3)})}{x^{1/3} (1 + x^{(1/5 - 1/3)})} \][/tex]
Notice that \(1/5\) is less than \(1/3\), so \(1/5 - 1/3\) is negative. Let’s simplify the exponent:
[tex]\[ \frac{x^{1/3} (1 - x^{-2/15})}{x^{1/3} (1 + x^{-2/15})} \][/tex]
### Step 3: Cancelling Out x^{1/3}
Now, cancel the common factor of \(x^{1/3}\) from both the numerator and the denominator:
[tex]\[ \frac{1 - x^{-2/15}}{1 + x^{-2/15}} \][/tex]
### Step 4: Evaluate the Limit
As \(x \to \infty\), \(x^{-2/15} \to 0\). Thus, the expression simplifies to:
[tex]\[ \frac{1 - 0}{1 + 0} = \frac{1}{1} = 1 \][/tex]
### Final Answer
Therefore, the limit is:
[tex]\[ \lim_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}} = 1 \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.