12.27

Workshop Week 11 - December 27, 2024

1. Compute the following limits:

(a) \(\lim_{x \to \infty} \left(e^x + x\right)^{\frac{1}{x}}\)

We can consider \(\ln\left(\lim_{x\to\infty}\left(e^{x}+x\right)^{\frac{1}{x}}\right)=\ln A\)

Or

\(\lim_{x\to\infty}\left(e^{x}+x\right)^{\frac{1}{x}}=\lim_{x\to\infty}e^{\frac{\ln\left(e^{x}+x\right)}{x}}=e^{\lim_{x\to\infty}\frac{\ln\left(e^{x}+x\right)}{x}}\)

Since \(\lim_{x\to\infty}\frac{\ln\left(e^{x}+x\right)}{x}=\lim_{x\to\infty}\frac{\frac{e^{x}+1}{e^{x}+x}}{1}=\lim_{x\to\infty}\frac{e^{x}}{e^{x}+1}=1\), then \(\lim_{x\to\infty}\left(e^{x}+x\right)^{\frac{1}{x}}=e\)​

(b) \(\lim_{x \to \infty} \left(\ln(x) - x\right)\)

\(\lim_{x\to\infty}\left(\ln(x)-x\right)=\lim_{x\to\infty}\left(\ln(x)-\ln\left(e^{x}\right)\right)=\lim_{x\to\infty}\left(\ln(\frac{x}{e^{x}})\right)=\ln(\lim_{x\to\infty}\frac{x}{e^{x}})=1\)

(c) \(\lim_{x\to0^{+}}\left(\sin(x)\right)^{\sqrt{x}}\)

\(\lim_{x\to0^{+}}\left(\sin(x)\right)^{\sqrt{x}}\)

(d) \(\lim_{x \to \infty} \left(3^x + 5^x\right)^{\frac{1}{x}}\)

(e) \(\lim_{x \to \frac{\pi}{2}^-} \tan(x)^{\cos(x)}\)

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2. Sketch the graphs of the following functions after finding:

• Domain
• Intervals of increase/decrease
• Intervals of concavity/convexity
• Points of local minima/local maxima
• Points of inflection
• Asymptotes

(a) \(f(x) = x^2 |x - 3|\)

(b) \(f(x) = \frac{x^2}{x^2 + 1}\)

(c) \(f(x) = 3x^4 - 8x^3 + 12\)

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3. Let \(f : \mathbb{R} \to \mathbb{R}\) be a twice-differentiable function with the following properties:

• \(f(-1) = 4\), \(f(0) = 2\), \(f(1) = 0\)
• \(f'(x) > 0\) for \(|x| > 1\)
• \(f'(x) < 0\) for \(|x| < 1\)
• \(f'(1) = f'(-1) = 0\)
• \(f''(x) < 0\) for \(x < 0\)
• \(f''(x) > 0\) for \(x > 0\)

Sketch the graph of \(f\).