Limits of Inverse Trigonometric Functions | Calculus

TL;DR

Inverse trigonometric functions have specific limits as x approaches positive or negative infinity, and it is helpful to understand the graph of these functions to evaluate the limits accurately.

Transcript

consider this problem what is the limit as x approaches infinity of the function that is the inverse tangent of x how can we find the answer it helps to know what the graph of inverse tan or arc tangent of x looks like there's two horizontal asymptotes the first one is y equals pi over two and the second horizontal asymptote is y equals negative pi... Read More

Key Insights

• ☺️ The limit as x approaches positive infinity of arc tangent of x is pi/2, and as x approaches negative infinity, it is -pi/2.
• 📈 Evaluating limits of inverse trigonometric functions requires knowledge of the shape of the graph.
• ⛔ Limits of inverse trigonometric functions can be different from the limits of the corresponding trigonometric functions.
• ⛔ In some cases, the left-sided and right-sided limits of an inverse trigonometric function can be different, indicating that the overall limit does not exist.
• 👨‍💼 The graph of arc sine has a limited domain and range, and the domain of arc sine is different from the domain of sine.
• 📈 Understanding the behavior of the graph of inverse trigonometric functions can help in evaluating limits accurately.
• ⛔ Multiplying the numerator and denominator of a rational function by 1/x^n can simplify the evaluation of limits at infinity.

Questions & Answers

Q: What is the limit as x approaches positive infinity of arc tangent of x?

The limit as x approaches positive infinity of arc tangent of x is pi/2. As x increases, the graph of arc tangent of x approaches the y-value of pi/2.

Q: What is the limit as x approaches negative infinity of arc tangent of x?

The limit as x approaches negative infinity of arc tangent of x is -pi/2. As x decreases, the graph of arc tangent of x approaches the y-value of -pi/2.

Q: What is the limit as x approaches 2 of arc tangent of x?

The limit as x approaches 2 of arc tangent of x does not exist. The left-sided and right-sided limits are different, indicating that the limit overall does not exist.

Q: How can we evaluate limits of inverse trigonometric functions?

To evaluate limits of inverse trigonometric functions, it is helpful to know the graph of the function. Understanding the behavior of the graph, such as the location of horizontal asymptotes and the shape of the curve, can assist in determining the limits.

Summary & Key Takeaways

• The limit as x approaches positive infinity of arc tangent of x is pi/2, as the graph of arc tangent of x approaches the y-value of pi/2 as x increases.

• The limit as x approaches negative infinity of arc tangent of x is -pi/2, as the graph of arc tangent of x approaches the y-value of -pi/2 as x decreases.

• The limit as x approaches 2 of arc tangent of x does not exist, as the right-sided and left-sided limits are different.