Find the limit. lim t-3 √/3-ti + In(t)j-1k)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Find the limit
Lim t->3 ( square root 3 - t i + ln(t) - 1/t k)
![### Limit Calculation
To solve the problem, find the limit as \( t \) approaches 3 for the given vector-valued function:
\[
\lim_{{t \to 3}} \left( \sqrt{3} - ti + \ln(t) j - \frac{1}{t} k \right)
\]
Breaking down the limit into its components and finding each separately:
1. **For the \(i\)-component:**
\[
\sqrt{3} - t
\]
As \( t \) approaches 3:
\[
\sqrt{3} - 3
\]
2. **For the \(j\)-component:**
\[
\ln(t)
\]
As \( t \) approaches 3:
\[
\ln(3)
\]
3. **For the \(k\)-component:**
\[
-\frac{1}{t}
\]
As \( t \) approaches 3:
\[
-\frac{1}{3}
\]
### Combined Limit
Combining the results from each component:
\[
\lim_{{t \to 3}} \left( \sqrt{3} - ti + \ln(t) j - \frac{1}{t} k \right)
= \left( \sqrt{3} - 3 \right) i + (\ln(3)) j - \left( \frac{1}{3} \right) k
\]
### Final Result:
\[
-3i + \ln(3) j - \frac{1}{3} k
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fc33cc1-e822-44c8-a303-0b5fd2ff9f49%2F63660059-66a0-4d66-8538-327336de5416%2F82t5bu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Limit Calculation
To solve the problem, find the limit as \( t \) approaches 3 for the given vector-valued function:
\[
\lim_{{t \to 3}} \left( \sqrt{3} - ti + \ln(t) j - \frac{1}{t} k \right)
\]
Breaking down the limit into its components and finding each separately:
1. **For the \(i\)-component:**
\[
\sqrt{3} - t
\]
As \( t \) approaches 3:
\[
\sqrt{3} - 3
\]
2. **For the \(j\)-component:**
\[
\ln(t)
\]
As \( t \) approaches 3:
\[
\ln(3)
\]
3. **For the \(k\)-component:**
\[
-\frac{1}{t}
\]
As \( t \) approaches 3:
\[
-\frac{1}{3}
\]
### Combined Limit
Combining the results from each component:
\[
\lim_{{t \to 3}} \left( \sqrt{3} - ti + \ln(t) j - \frac{1}{t} k \right)
= \left( \sqrt{3} - 3 \right) i + (\ln(3)) j - \left( \frac{1}{3} \right) k
\]
### Final Result:
\[
-3i + \ln(3) j - \frac{1}{3} k
\]
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