Evaluate the following limits. limx →π/ 2^- (tanx)/(3 /(2 x-π))

Name: Problem 39, Chapter 4: Calculus Early Transcendentals
Uploaded: 2021-08-12T00:54:59-08:00
Duration: 5 min 20 s
Channel: Christy Galilei
Description: Evaluate the following limits. limx →π/ 2^- (tanx)/(3 /(2 x-π))

Evaluate the following limits. limx →π/ 2^- (tanx)/(3 /(2...

Key Concepts

Series Expansion/First-Order Approximation

Series expansion techniques, such as the Taylor Series, allow functions to be approximated by polynomials near a given point. First-order approximations are particularly useful in limit evaluations because they simplify complex functions into linear forms, revealing the dominant behavior as the variable approaches a specific value.

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a specified value. They are essential for understanding continuity, derivatives, and integrals, and they allow us to analyze the behavior of functions even when they are not explicitly defined at certain points.

Indeterminate Forms

Indeterminate forms arise in limit evaluations when direct substitution results in expressions like 0/0 or ?/?, which do not provide clear information about the limit. Recognizing these forms is crucial because they signal the need for further methods, such as algebraic manipulation or advanced techniques, to evaluate the limit accurately.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms. It involves differentiating the numerator and the denominator independently and then re-evaluating the limit. This rule is a powerful tool in calculus for simplifying and resolving limits that are not directly computable.

Trigonometric Function Behavior Near Singularities

Trigonometric functions often exhibit rapid changes or asymptotic behavior near points where they are undefined or tend toward infinity. Understanding the behavior of these functions near their singularities is crucial for evaluating limits involving trigonometric expressions, as it helps in applying appropriate approximations and transformations.

Transcript

00:14 Okay, we are told in the directions for this problem that this is the indeterminate form of infinity over infinity.

00:20 So we will use lebetal's rule.

00:27 So the derivative of the numerator, tangent, is sikin squared.

00:35 And the denominator derivative, we can use the quotient rule.

00:42 So the quotient rule is low, the denominator.

00:46 Oops, that's 2x.

00:51 D high derivative of the numerator 0 minus high, d low derivative of the denominator is 2 all over the denominator squared.

01:05 All right, so i'd like to simplify this, and we have sequence squared numerator divided by, and then i'm going to write the denominator.

01:13 So that first part is nothing, so that's negative 6 over 2x minus pi squared.

01:21 Now, i'm going to rewrite sikin.

01:23 Seekin is the reciprocal of cosine.

01:26 So i'm going to make this 1 over cosine x squared, which i'm going to write like that.

01:32 Okay, then we can divide two fractions by flipping the second one over and multiplying.

01:47 So that's going to give us 2x minus pi squared over negative 6 times cosine x squared.

01:57 And that is the limit as x approaches pi over 2 from the left.

02:05 So let's plug this in.

02:08 So we have 2 times pi over 2 minus pi squared.

02:17 So 2 times pi over 2 with pi, pi minus pi squared is 0.

02:22 Over the denominator, negative 6 times the cosine of pi over 2 squared.

02:28 So the cosine of pi over 2 is 0.

02:32 So we get 0 over 0, and we have to use the local 12 again.

02:38 So we're doing the derivative right here...

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