Find the limit. limx →+∞(1-3 / x)^x

Name: Problem 25, Chapter 4: Calculus A New Horizon
Uploaded: 2021-07-02T05:46:02-08:00
Duration: 3 min 43 s
Channel: Manisha Sarker
Description: Find the limit. limx →+∞(1-3 / x)^x

Key Concepts

Limits at Infinity

This concept deals with finding the value that a function approaches as the independent variable grows without bound. It is fundamental in calculus for understanding the long-term behavior of functions and is used to analyze series, asymptotic behavior, and stability in various mathematical contexts.

Exponential Limit Forms

Exponential limit forms refer to limits involving expressions of the form (1 + c/x)^x or similar variations. These forms are important because they often converge to values involving the natural exponential function. Recognizing such patterns allows one to apply known limit results, such as the limit definition of the exponential function.

Definition of the Natural Exponential Constant (e)

The number e is defined by the limit of the form (1 + 1/n)^n as n tends to infinity. More generally, limits of the form (1 + c/x)^x can be expressed in terms of e by rewriting them as e^c. This definition is crucial in calculus and analysis for its natural occurrence in growth processes, differential equations, and continuous compounding problems.

Indeterminate Forms

Indeterminate forms, like 1^?, occur when the limit of a function yields an uncertain expression that cannot be directly evaluated. Recognizing and correctly handling these forms using logarithmic transformation or other analytical techniques is key to determining the true behavior of the function as the variable approaches a limit.

Transcript

00:01 In this problem, we need to apply a l -opetals rule in order to determine a given limit.

00:05 Now, let us consider y to be equal to 1 minus 3 by x to the power x, because we need to determine the limit as x tends to infinity of this expression.

00:18 So let us take the natural logarithm on both sides.

00:20 So we have that lawn -y is equal to lawn of this expression.

00:23 And using the power rule of logarithms, this will be equal to x times loan of 1 minus 3 by x.

00:32 So we can consider the limit in this case.

00:38 That is consider the limit as x tends to plus infinity of y.

00:44 Then that will be equal to the limit as x tends to plus infinity of x times lawn of 1 minus.

00:54 3 by x.

00:56 Now note that on the right hand side we have x tends to infinities and then we have lawn of 1 minus 3 by x as x tends to infinity that will be 1 minus 0.

01:06 So just 1 and lawn of 1 is 0.

01:09 So we have an infinity time 0 form and we cannot apply a law.

01:12 It is rule like this.

01:14 So we rewrite this as limit x tends to plus infinity of lawn of 1 minus 3 by x divided by 1 divided by x.

01:22 Then the numerator will tend to 0 and the denominator will also tend to 0 as x tends to plus infinity and we have a 0 by 0 form and thus we can apply l hoppetals rule.

01:33 So applying l hoppetals rule, we take the derivative of the numerator.

01:38 So that will be 1 divided by 1 minus 3 by x times the derivative of 1 minus 3 by x.

01:44 So that will be 0 minus 3 times minus 1 by x square.

01:49 And in the denominator, we have the derivative of 1 by x.

01:53 So that will be minus 1 by x squared...

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