limit of tan^-1(1/(x-4)) as x goes to 4, calculus 1 tutorial

TL;DR

How to calculate the limit as x approaches 4 of inverse tangent of 1 over x minus 4.

Transcript

okay i'll show you guys how to calculate this  limit the limit as x approaching 4 of inverse   tangent of 1 over x minus 4. first of all this  right here is just the inverse tangent notation   and of course sometimes we can write it  as arc the arc the arc tangent both of   them mean the same thing anyway suppose we're  just plugging this 4 into th... Read More

Key Insights

• ⛔ The limit of the inverse tangent function can be evaluated by considering the positive and negative directions of the input.
• ⛔ When approaching 4 from the positive direction, the limit tends to positive infinity.
• ❎ When approaching 4 from the negative direction, the limit tends to negative infinity.
• ⛔ The discrepancy between the two directions results in the limit not existing.
• 😀 The graph of the inverse tangent function has two horizontal asymptotes at y = pi/2 and y = -pi/2.
• ♾️ Inverse tangent of infinity corresponds to the y value of pi/2 on the graph.

Questions & Answers

Q: How do you calculate the limit as x approaches 4 of the inverse tangent of 1 over x minus 4?

To calculate this limit, we need to consider the positive and negative directions. When approaching 4 from the positive direction, the limit tends to positive infinity. When approaching 4 from the negative direction, it tends to negative infinity. Therefore, the limit does not exist.

Q: What does it mean when a limit does not exist?

When a limit does not exist, it means that the function does not approach a specific value as the input approaches a certain value. In this case, as x approaches 4, the limit does not converge to a finite value.

Q: How is the limit from the positive direction calculated?

To calculate the limit from the positive direction, we substitute a number slightly larger than 4 into the function. In this case, when plugging in 4.0001, the resulting expression simplifies to the inverse tangent of 1 over 0 plus, which tends to positive infinity.

Q: What is the limit from the negative direction?

The limit from the negative direction is calculated by substituting a number slightly smaller than 4 into the function. When plugging in 3.999, the resulting expression simplifies to the inverse tangent of 1 over 0 minus, which tends to negative infinity.

Summary & Key Takeaways

• The content explains how to calculate the limit as x approaches 4 of the inverse tangent of 1 over x minus 4.

• The limit is evaluated by breaking it into two pieces - one from the positive direction and one from the negative direction.

• When approaching 4 from the positive direction, the result tends to positive infinity, and when approaching 4 from the negative direction, it tends to negative infinity. Therefore, the limit does not exist.