Exploring the Limit of x / e^(-1/x) as x Approaches Zero

In mathematics, understanding limits is crucial for solving many problems in calculus and analysis. One specific limit that is particularly interesting is the limit of x1e1x as x approaches zero. This exploration will help us understand the behavior of the function as x gets very close to zero, both from the left and the right.

Definition of the Limit

Mathematically, the limit of a function f(x) as x approaches a certain value, say a, is the value that f(x) approaches as x gets arbitrarily close to a.

Analysis from the Right

When approaching zero from the right (x → 0 ), the expression x1e1x can be broken down as follows:

x1e1x as xrarr;0 1e1x approaches ∞ as xrarr;0 1e1x can be rewritten as 1/e1x, thus the expression becomes x1/e1x Which simplifies to xe1x

As xrarr;0 , the numerator x approaches 0 and the denominator e1x approaches ∞. By the property of infinity, the overall expression approaches 0.

Analysis from the Left

Similarly, when approaching zero from the left (x → 0-), the expression can be analyzed as:

x1e1x as xrarr;0- 1e1x approaches 1∞0 as xrarr;0- The expression becomes x0 , which also approaches 0.

Conclusion

Based on the above analysis, the limit of x1e1x as x approaches 0 from either direction is 0. Therefore, the limit is:

x1e1x as xrarr;0, the limit is 0.

Proof Using Indeterminate Form

When dealing with limits involving indeterminate forms like 0/0 or ∞/∞, we can use L'H?pital's rule. In this case, we can separate the limit into the product of x and 1/e^(1/x). Let:

limxx#x2192; and limxx#x2192;01e1x1x

Both of these individual limits approach the form 0/0, and thus the product of the limits also approaches 0. This confirms our previous analysis.

Graphical Validation

To further validate the limit, we can use a graphing tool like Desmos. By plotting the function y x / (1/e^(1/x)), we can visually see that the function approaches 0 as x approaches 0 from both directions.