Solving the Limit L lim_{x rightarrow infty}left(frac{2x^3}{x^1}right)^{2x-1}

In this article, we will walk you through the detailed process of solving the limit

[ L lim_{x rightarrow infty}left(frac{2x^3}{x^1}right)^{2x-1} ]

Simplify the Expression Inside the Limit

First, we start by simplifying the expression inside the limit. The expression is

[ frac{2x^3}{x^1} ]

As (x) approaches infinity, we can simplify this fraction as follows:

[ frac{2x^3}{x^1} frac{2 frac{3}{x}}{1 frac{1}{x}} ]

As (x to infty), both (frac{3}{x}) and (frac{1}{x}) approach 0. Therefore,

[ frac{2x^3}{x^1} to frac{2 cdot 0}{1 cdot 0} 2 ]

Examine the Limit of the Exponent

Now we rewrite the limit using the simplified expression:

[ L lim_{x to infty} left(2right)^{2x-1} ]

We can simplify this further using the property of exponents:

[ 2^{2x-1} 2^{2x} cdot 2^{-1} frac{2^{2x}}{2} ]

As (x to infty), (2^{2x}) approaches infinity. Therefore,

[ L lim_{x to infty} frac{2^{2x}}{2} infty ]

Thus, the limit is boxed{infty}.

Further Explanation and Substitution

First, let's calculate the limit within the parenthesis:

[ lim_{x to infty} frac{2x^3}{x^1} 2 ]

So, the original limit is equivalent to:

[ lim_{x to infty} left(frac{2x^3}{x^1}right)^{2x-1} lim_{x to infty} 2^{2x-1} infty ]

This shows that as (x) approaches infinity, the expression (2^{2x-1}) also approaches infinity.

Right and Left Limits

Considering the behavior of the function from both the right and left sides of infinity:

[ lim_{x to infty^{-}} left(frac{2x^3}{x^1}right)^{2x-1} ]

We can perform the same simplification process:

[ lim_{x to infty^{-}} left(frac{2x^3}{x^1}right)^{2x-1} lim_{x to infty^{-}} frac{2x^3}{x^1}^{2x} lim_{x to infty^{-}} left(frac{2tfrac{3}{x}}{1tfrac{1}{x}}right)^{2x} lim_{x to infty^{-}} left(frac{2 cdot 0}{1 cdot 0}right)^{2x} 0 ]

Now, consider the limit as (x) approaches infinity from the right side:

[ lim_{x to infty^{ }} left(frac{2x^3}{x^1}right)^{2x-1} lim_{x to infty^{ }} left(frac{2tfrac{3}{x}}{1tfrac{1}{x}}right)^{2x} lim_{x to infty^{ }} left(frac{2 cdot 0}{1 cdot 0}right)^{2x} infty ]

Since the left limit is 0 and the right limit is infty, the overall limit does not exist.

Conclusion

In conclusion, the limit L lim_{x rightarrow infty}left(frac{2x^3}{x^1}right)^{2x-1} does not exist due to the discontinuity in the behavior of the function as x approaches infinity from both sides.

Keywords

limit calculation, infinity limit, exponent limit

References

Wikipedia - Limit of a Function

Math is Fun - Limits