solve this problem---Question:
Find the limit:
$\lim_{x \to +\infty} (\sqrt{x^2 + 2x} - x)$
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rednote ID: 2968362456
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Let's find the limit of square root of x squared plus 2x minus x as x approaches positive infinity. When we try direct substitution, we get infinity minus infinity, which is an indeterminate form. This graph shows how the function approaches the value 1 as x increases. We need special algebraic techniques to solve this type of limit involving square roots and infinity.
To solve this indeterminate form, we use the rationalization technique. We multiply and divide by the conjugate expression: square root of x squared plus 2x plus x. This eliminates the square root in the numerator using the difference of squares formula: a minus b times a plus b equals a squared minus b squared. Here, a is the square root term and b is x.
Now we work through the algebraic simplification after rationalization. The numerator becomes x squared plus 2x minus x squared, which simplifies to 2x. The denominator remains square root of x squared plus 2x plus x. This gives us the simplified form: 2x over square root of x squared plus 2x plus x. This new form is much more manageable for taking the limit as x approaches positive infinity.
Now we factor out x from both numerator and denominator. We get 2x over x times the square root of 1 plus 2 over x plus 1. This simplifies to 2 over square root of 1 plus 2 over x plus 1. As x approaches positive infinity, 2 over x approaches 0, so the square root of 1 plus 2 over x approaches 1. Therefore, the limit equals 2 over 1 plus 1, which equals 1.
Let's summarize our complete solution. We started with the limit of square root of x squared plus 2x minus x as x approaches positive infinity. Using rationalization, we transformed it to 2x over square root of x squared plus 2x plus x. After factoring out x, we got 2 over square root of 1 plus 2 over x plus 1, which equals 1. We can verify this because for large x, square root of x squared plus 2x is approximately x plus 1. The key techniques we used were identifying indeterminate forms, rationalization with conjugates, algebraic simplification, and factoring for limit evaluation.