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February 1, 2025 at 4:56 am #1200
Rajeev Bagra
KeymasterHelp regarding computation of delta value
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Understanding the Computation of (\delta)
If you’re solving for (\delta) using the (\epsilon)-(\delta) definition of a limit, you might notice that solving for ( f(x) = 1.95 ) and ( f(x) = 2.05 ) gives three values for ( x ). Since the equation is cubic (( x^3 – 4x + 5 )), it’s expected to have up to three real roots.
How to Determine (\delta):
- Find the solutions for:
[latex]\[ f(x) = 1.95 \quad \text{and} \quad f(x) = 2.05 \][/latex]
- Identify the closest solutions to ( x = 1 ) (since we analyze limits around ( x = 1 )).
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Compute (\delta) as half the distance between these closest solutions:
[latex]\[ \delta = \frac{|x_{\text{upper}} - x_{\text{lower}}|}{2} \][/latex]
where ( x_{\text{upper}} ) and ( x_{\text{lower}} ) are the closest solutions from the two sets.
Why Are There Three Values?
- Since ( f(x) ) is cubic, it intersects the horizontal lines ( y = 1.95 ) and ( y = 2.05 ) at three points.
- However, for (\delta), we only care about the values closest to ( x = 1 ).
- The other solutions further from ( x = 1 ) do not contribute to (\delta) in the (\epsilon)-(\delta) definition.
Final Check:
✅ If your graphical solution confirms that the two closest solutions form a symmetric interval around ( x = 1 ), then your (\delta) computation is correct.
⚠️ If the solutions are asymmetric, then you should take the minimum deviation from ( x = 1 ) instead of half the full range.
<p><strong>Solutions for ( f(x) = 1.95 ):</strong></p>
[latex]\[
x \approx -2.30696, \quad x \approx 1.06163, \quad x \approx 1.24534
\][/latex]<p><strong>Solutions for ( f(x) = 2.05 ):</strong></p>
[latex]\[
x \approx -2.29857, \quad x \approx 0.95578, \quad x \approx 1.34279
\][/latex]<p><strong>Closest solutions to ( x = 1 ) (used for delta computation):</strong></p>
[latex]\[
\text{Lower bound}: x \approx 1.06163 \quad \text{(from \( f(x) = 1.95 \))}
\][/latex]
[latex]\[
\text{Upper bound}: x \approx 0.95578 \quad \text{(from \( f(x) = 2.05 \))}
\][/latex]<p><strong>Computed ( \delta ) value:</strong></p>
[latex]\[
\delta \approx 0.05292
\][/latex] -
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