The limit as x approaches infinity

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SUMMARY

The limit of the expression (3x - 2) / (9x + 7) as x approaches infinity can be evaluated by factoring out x from both the numerator and denominator. This results in the simplified form of (3 - 2/x) / (9 + 7/x). As x approaches infinity, the terms 2/x and 7/x approach zero, leading to a final limit of 3/9, which simplifies to 1/3. The discussion emphasizes that direct substitution of infinity is not valid, and proper limit evaluation techniques must be applied.

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Homework Statement


(3x-2) / (9x+7)

As x approaches infinity.

The Attempt at a Solution


I know the procedure, but am then stuck:
- Rearrange
- Plug in infinity for x
- Evaluate

Tried breaking it into 3x / (9x+7) - 2 / (9x+7)
The second thing would go to zero.
The first, no clue.

Maybe multiplying the original problem by something? Unsure what.

(Note: This is to help a friend. I myself am going to bed now, but they will be checking for replies. They're not allowed to use L'Hopital's rule, as it wasn't taught)
 
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factor out x from both num and den!
and plug in the infinity value without thinking
 
Goldenwind said:

Homework Statement


(3x-2) / (9x+7)

As x approaches infinity.

The Attempt at a Solution


I know the procedure, but am then stuck:
- Rearrange
- Plug in infinity for x
- Evaluate
No, you don't "know the procedure". Only in the most trivial limits (limits of continuous functions where, by definition, the limit is the value of the function, can)you "plug" the target point into the formula. And certainly you can't "plug infinity" into this formula because it is only defined for real numbers and "infinity" is not a real number.

Tried breaking it into 3x / (9x+7) - 2 / (9x+7)
The second thing would go to zero.
The first, no clue.

Maybe multiplying the original problem by something? Unsure what.

(Note: This is to help a friend. I myself am going to bed now, but they will be checking for replies. They're not allowed to use L'Hopital's rule, as it wasn't taught)
It is true, however, that limit, as "x goes to infinity", of 1/x is 0 and 0 is easy to use. In order to change from "x" to "1/x" divide both numerator and denominator by x (that is the same as rootX's "factor x from numerator and denominator").
 
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