Calculating the Limit of ( frac{4x^5}{sqrt{3x^2 - 2}} ) as ( x ) Approaches Infinity

In calculus, determining limits as ( x ) approaches infinity is a fundamental concept. One common method involves simplifying the expression by dividing both the numerator and the denominator by the highest power of ( x ) present. This technique is especially useful in oscillating functions and functions involving square roots.

Step-by-Step Calculation

Let's find the limit of the expression ( frac{4x^5}{sqrt{3x^2 - 2}} ) as ( x ) approaches infinity.

First, rewrite the expression by dividing both the numerator and the denominator by ( x^5 ), the highest power of ( x ) in the denominator:

[ frac{4x^5}{sqrt{3x^2 - 2}} frac{4 cdot frac{x^5}{x^5}}{sqrt{frac{3x^2}{x^5} - frac{2}{x^5}}} frac{4}{sqrt{frac{3x^2 - 2}{x^5}}} frac{4}{sqrt{frac{3}{x^3} - frac{2}{x^5}}} ]

Now, as ( x ) approaches infinity, both ( frac{3}{x^3} ) and ( frac{2}{x^5} ) approach 0. Therefore, the expression simplifies to:

[ lim_{x to infty} frac{4}{sqrt{frac{3}{x^3} - frac{2}{x^5}}} frac{4}{sqrt{0 - 0}} frac{4}{sqrt{0}} ]

However, we can rationalize the denominator to get:

[ frac{4}{sqrt{3}} cdot frac{sqrt{3}}{sqrt{3}} frac{4sqrt{3}}{3} ]

Thus, the limit is ( frac{4sqrt{3}}{3} ).

Verification Using Specific Values

To verify this, let's substitute some specific large values for ( x ) into the original expression and observe the trend:

When ( x 100 ), ( 4x^5 5 405 ) and ( sqrt{3x^2 - 2} sqrt{29998} approx 173.199 ). The fraction is approximately ( frac{405}{173.199} approx 2.33 ). When ( x 200 ), ( 4x^5 5 805 ) and ( sqrt{3x^2 - 2} sqrt{119998} approx 346.407 ). The fraction is approximately ( frac{805}{346.407} approx 2.323 ). When ( x 300 ), ( 4x^5 5 1205 ) and ( sqrt{3x^2 - 2} sqrt{269998} approx 519.613 ). The fraction is approximately ( frac{1205}{519.613} approx 2.311 ). When ( x 400 ), ( 4x^5 5 1605 ) and ( sqrt{3x^2 - 2} sqrt{479998} approx 692.819 ). The fraction is approximately ( frac{1605}{692.819} approx 2.327 ).

As ( x ) increases, the value of the fraction approaches ( frac{4}{sqrt{3}} approx 2.309 ).

Conclusion

In conclusion, the limit of the expression ( frac{4x^5}{sqrt{3x^2 - 2}} ) as ( x ) approaches infinity is ( frac{4sqrt{3}}{3} ).

This method of analysis and verification using specific values helps us understand the behavior of the function at very large values of ( x ).