Limit of x*sin(1/x) as x approaches infinity || Two Solutions

TL;DR

Limit as x approaches infinity of x times sine(1/x) is equal to 1 using substitution or L'Hopital's rule.

Transcript

hi in this video we're going to try to find the limit so we have the limit as x approaches infinity of x times the sine of one over x so what i'm thinking about doing in this problem and i haven't done this yet is basically do the following so this is the limit as x approaches infinity and then i'm going to write it like this sine of 1 over x over ... Read More

Key Insights

• ☺️ Substituting t=1/x allows the limit of x times sine(1/x) to be simplified to a known limit form.
• ☺️ L'Hopital's rule is utilized to find the limit of cosine(1/x) as x approaches infinity.
• ☺️ The limit of x times sine(1/x) can be computed in multiple ways, showcasing the versatility of calculus techniques.

Questions & Answers

Q: How can the limit of x times sine(1/x) as x approaches infinity be found?

The limit can be found by substituting t=1/x and applying the known limit of sine t/t as t approaches zero, simplifying to 1.

Q: What is L'Hopital's rule and how is it applied in finding the limit?

L'Hopital's rule states that for 0/0 forms, the limit can be found by taking the derivative of the numerator and denominator separately. This rule is used to find the limit of cosine(1/x) as x approaches infinity.

Q: Why does the limit of x times sine(1/x) simplify to cosine(0) which equals 1?

As x approaches infinity, 1/x approaches zero, making sine(1/x)/1/x equivalent to sine t/t as t approaches zero, resulting in the limit being 1.

Q: What are the two different approaches to finding the limit discussed in the video?

The two methods shown are using substitution of t=1/x and applying L'Hopital's rule for 0/0 forms to compute the limit as x approaches infinity of x times sine(1/x).

Summary & Key Takeaways

• The video discusses finding the limit as x approaches infinity of x times sine(1/x).

• Two methods are demonstrated: using the substitution t=1/x and applying L'Hopital's rule for 0/0 forms.

• The limit simplifies to cosine(0) which is 1, providing two solutions to the problem.