Chapter 4, Problem 14RE

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Evaluate the limit. lim x → ( π / 2 ) − ( tan x ) cos x

To determine

To evaluate: The value of limx(π2)(tanx)cosx .

Explanation

Given:

The function is, limxâ†’(Ï€2)âˆ’(tanx)cosx .

Calculation:

Obtain the value of the function as x approaches Ï€2 .

limxâ†’(Ï€2)âˆ’(tanx)cosx=(tan(Ï€2))cos(Ï€2)=âˆž0

Thus, limxâ†’(Ï€2)âˆ’(tanx)cosx=âˆž0 is in an indeterminate form.

Therefore, apply Lâ€™Hospitalâ€™s Rule and obtain the limit as shown below.

Let the given function as y=(tanx)cosx .

Take logarithm on both sides,

lny=ln(tanx)cosx=cosxln(tanx)=ln(tanx)secx

Apply the limit and obtain the value.

limxâ†’(Ï€2)âˆ’lny=limxâ†’(Ï€2)âˆ’ln(tanx)secxlny=ln(tan(Ï€2))sec(Ï€2)lny=ln(tan(Ï€2))sec(Ï€2)=undefined

Again it is in indeterminate form

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