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Answer :
Final answer:
The limit of the given function as x approaches infinity is found by simplifying the expression and observing that the terms with x in the denominator approach zero. This leave us with the simplified terms which yield the limit of 3/5 or 0.6.
Explanation:
Understanding Limits in Mathematics
To solve limit problems, we employ strategies based on algebraic simplification and the properties of limits. When calculating the limit as x approaches infinity, terms with lower degrees become negligible compared to terms with higher degrees, as they don't impact the value of the limit significantly.
For the given function f(x) = 3x² + 3 over 5x² + 7x - 39, first we factor out x² from both the numerator and denominator to simplify the limit:
The simplified denominator becomes 5 + 7/x - 39/x².
As x approaches infinity, the terms 3/x², 7/x, and 39/x² all approach 0 since their denominators grow without bound.
limx→∞ (3 + 3/x²) which simplifies to 3 as x approaches infinity.
limx→∞ (5 + 7/x - 39/x²) which simplifies to 5 as x approaches infinity.
The final calculated limit is the ratio of these two simplified terms, which yields 3/5 or 0.6.
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