Question
19. [-1t Points] DETALS MY NOTES SCALCETY 2XP2OTL. ASK FOUR Use a tablie of values the estimate the value of the limit. If you have a graphing ctevics, use it un anmimn your tesut graghicail. (Round your answer to bwo decimal places.) \[ \lim _{x \rightarrow \infty} \frac{\sqrt{x+36}-6}{x} \] \( \square \) Need Help? Beodit Wamel t
Ask by Cole Carrillo. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limit is 0.00.
Solution
To evaluate the limit
limₓ→∞ [(√(x + 36) − 6) / x],
we can start by simplifying the expression.
Step 1. Rationalize the numerator:
Multiply the numerator and denominator by (√(x + 36) + 6):
(√(x + 36) − 6) * (√(x + 36) + 6) = (x + 36) − 36 = x
Thus, the original expression becomes:
[(x) / (√(x + 36) + 6)] / x = 1 / (√(x + 36) + 6).
Step 2. Now take the limit:
As x → ∞, √(x + 36) → ∞, so the denominator √(x + 36) + 6 → ∞, which implies
limₓ→∞ 1 / (√(x + 36) + 6) = 0.
Step 3. Using a table of values (rounded to two decimal places):
x √(x + 36) − 6 (√(x + 36) - 6) / x
100 √136 ≈ 11.66, so 11.66 - 6 = 5.66 5.66 / 100 ≈ 0.06
1,000 √1036 ≈ 32.19, so 32.19 - 6 = 26.19 26.19 / 1,000 ≈ 0.03
10,000 √10036 ≈ 100.18, so 100.18 - 6 = 94.18 94.18 / 10,000 ≈ 0.0094
As x becomes larger, the value of (√(x + 36) − 6) / x gets closer and closer to 0.
Answer (rounded to two decimal places): 0.00.
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Simplify this solution Extra Insights
To estimate the limit \(\lim _{x \rightarrow \infty} \frac{\sqrt{x+36}-6}{x}\) using a table of values, pick large values of \(x\) such as 100, 1000, and 10,000. As you compute, you'll find that the expression approaches zero. The pattern indicates that as \(x\) increases, the limit will get closer to 0. For a more visual approach, you can graph the function \(\frac{\sqrt{x+36}-6}{x}\) on graphing software. As you observe the graph while extending \(x\) towards infinity, you'll see the curve quickly approaches the x-axis, further confirming that the limit converges to 0, providing a great visual representation of what happens at infinity!
