2.3: The Limit Laws
- Page ID
- 143299
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- Evaluate \(\displaystyle \lim_{x \to 2} 2x^2\), if possible.
- Evaluate \(\displaystyle \lim_{x \to -3} 7\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 5} (3x - 12)\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 2} \dfrac{x+7}{x-3}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1} \dfrac{\ln x}{\cos(\pi x)}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -1} \sqrt{x^2 - 7x - 2}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 0} \dfrac{x}{x}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -3} \dfrac{x^2 - 9}{x + 3}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 4} \dfrac{x^2 - 6x + 8}{x^2 + x - 20}\), if possible.
- Evaluate \( \displaystyle \lim_{h \to 0} \dfrac{(x + h)^2 - x^2}{h} \), if possible.
- Evaluate \( \displaystyle \lim_{t \to 4} \dfrac{\sqrt{t} - 2}{t - 4} \), if possible.
- Evaluate \( \displaystyle \lim_{\theta \to \pi} \dfrac{\sin \theta}{\tan \theta}\), if possible.
- Evaluate \(\displaystyle \lim_{h \to 0} \dfrac{\dfrac{1}{x+h} - \dfrac{1}{x}}{h}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 5} \dfrac{x - 5}{3 - \sqrt{x + 4}} \), if possible.
- Evaluate \(\displaystyle \lim_{x \to 0} \sin\left(\dfrac{1}{x}\right)\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -2^-} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -2^+} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -2} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 3^-} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 3^+} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 3} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1^-} \dfrac{1}{x^2 - 2x + 1}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1^+} \dfrac{1}{x^2 - 2x + 1}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1} \dfrac{1}{x^2 - 2x + 1}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 0^+} \sqrt{x}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 0^+} \ln x\), if possible.
- Consider the function
\[ f(x) = \left\{\begin{align*}
& x^2, & x < 1, \\
& x, & x \ge 1.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to 1^-} f(x)\), \(\displaystyle\lim_{x \to 1^+} f(x)\), and \(\displaystyle\lim_{x \to 1} f(x)\), if possible.
- Consider the function
\[ g(x) = \left\{\begin{align*}
& \cos x, & x \le 0, \\
& \sin x, & x > 0.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to 0^-} g(x)\), \(\displaystyle\lim_{x \to 0^+} g(x)\), and \(\displaystyle\lim_{x \to 0} g(x)\), if possible.
- Consider the function
\[ h(x) = \left\{\begin{align*}
& \dfrac{1}{x+1}, & x < -1, \\
& \dfrac{1}{x}, & -1 \le x < 0.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to -1^-} h(x)\), \(\displaystyle\lim_{x \to -1^+} h(x)\), and \(\displaystyle\lim_{x \to -1} h(x)\), if possible.
- Consider the function
\[ f(x) = \left\{\begin{align*}
& |1 - x^2|, & x < 0, \\
& \sqrt{2}, & x = 0, \\
& e^x, & x > 0.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to 0^-} f(x)\), \(\displaystyle\lim_{x \to 0^+} f(x)\), and \(\displaystyle\lim_{x \to 0} f(x)\), if possible.