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Phase 1: MACLRN
Phase 2: DIVERG
Phase 3: FINITE
Phase 4: TAYLOR
\[f(x) = m + \frac{\sin 3x}{x^3} + \frac{n}{x^2}, \qquad m,\, n \in \mathbb{R}\]
⏱ ~25 min👥 Groups of 3 · VNPS📐 Topic 5 · Calculus
Try substituting small values of \(x\), then confirm with the standard result.
\[\lim_{x \to 0} \frac{\sin 5x}{x} = \;?\]
What rule did you use?
What indeterminate form do you get? What technique can you use?
\[\lim_{x \to 0} \frac{\sin 3x}{x^3} = \;?\]
Does this limit exist?
For what value of \(n\) does the limit below exist as a finite number? Show your full reasoning — explain why only one value works.
\[\lim_{x \to 0} \left(\frac{\sin 3x}{x^3} + \frac{n}{x^2}\right)\]
Hint: expand \(\sin 3x\) as a Maclaurin series and divide by \(x^3\).
\[f(x) = m + \frac{\sin 3x}{x^3} + \frac{n}{x^2}, \qquad m,\,n \in \mathbb{R}\]
Show that a finite limit for \(\displaystyle\lim_{x \to 0} f(x)\) can only exist when \(n = -3\).
Using \(n = -3\) from part (a), find the value of \(m\) for which:
\[\lim_{x \to 0} f(x) = 5\]
Show every step of the calculation. Exact form required.
What changes? Find the new \(n\) and the new \(m\) if the limit still equals 5.
Solve part (b) using L'Hôpital's rule. How many applications? Which method is more efficient?
\[f(x) = m + \frac{\sin ax}{x^3} + \frac{n}{x^2}, \qquad a \in \mathbb{R}\]
Find \(n\) in terms of \(a\). Find \(\displaystyle\lim_{x\to 0} f(x)\) in terms of \(m\) and \(a\). If the limit equals \(L\), what is \(m\)?