2.2: The Limit of a Function
- Page ID
- 143298
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- Sketch the graph of a function \(f(x)\) that is continuous at \(x = 1\) with \(\displaystyle \lim_{x \to 1} f(x) = -2\).
- Sketch the graph of a function \(g(x)\) that is discontinuous at \(x = 1\) with \(\displaystyle \lim_{x \to 1} g(x) = -2\).
- Sketch the graph of a function \(h(x)\) with \(\displaystyle \lim_{x \to -2^-} h(x) = 3\), \(\displaystyle \lim_{x \to -2^+} h(x) = 1\), and \(h(-2) = 3\).
- Sketch the graph of a function \(f(t)\) with \(\displaystyle \lim_{t \to 0} f(t) = -\infty\).
- Sketch the graph of a function \(g(t)\) with \(\displaystyle \lim_{t \to -1^-} g(t) = \infty\), \(\displaystyle \lim_{t \to -1^+} g(t) = 0\), and \(g(1) = 2\).
- Given only that \(\displaystyle \lim_{x \to 11} h(x) = 4\) for some function \(h(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11^-} h(x)\) and \(\displaystyle \lim_{x \to 11^+} h(x)\)? Briefly explain why or why not.
- Given only that \(\displaystyle \lim_{x \to 11} f(x) = 4\) for some function \(f(x)\), is it possible to calculate \(f(11)\)? Briefly explain why or why not.
- Given only that \(g(11) = 4\) for some function \(g(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} g(x)\)? Briefly explain why or why not.
- Given only that \(h(11) = 4\) for some continuous function \(h(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} h(x)\)? Briefly explain why or why not.
- Given only that \(\displaystyle \lim_{x \to 11^-} f(x) = 4\) and \(\displaystyle \lim_{x \to 11^+} f(x) = 4\) for some function \(f(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} f(x)\)? Briefly explain why or why not.
- Given only that \(\displaystyle \lim_{x \to 11^-} f(x) = 5\) and \(\displaystyle \lim_{x \to 11^+} f(x) = 2\) for some function \(f(x)\), is it possible to calculate \(\displaystyle \lim_{x \to 11} f(x)\)? Briefly explain why or why not.
- Use a calculator to complete the table below for the function \(g(x) = 2x + 1\), then use your results to make inferences about \(\displaystyle \lim_{x \to -1^-} g(x)\), \(\displaystyle \lim_{x \to -1^+} g(x)\), and \(\displaystyle \lim_{x \to -1} g(x)\).
\(x\) \(g(x)\) \(x\) \(g(x)\) -1.01 -0.99 -1.001 -0.999 -1.0001 -0.9999
- Use a calculator to complete the table below for the function \(h(x) = \dfrac{\sin(x)}{x}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 0^-} h(x)\), \(\displaystyle \lim_{x \to 0^+} h(x)\), and \(\displaystyle \lim_{x \to 0} h(x)\).
\(x\) \(h(x)\) \(x\) \(h(x)\) -0.01 0.01 -0.001 0.001 -0.0001 0.0001
- Use a calculator to complete the table below for the function \(f(x) = \dfrac{x^2-1)}{|x-1|}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 1^-} f(x)\), \(\displaystyle \lim_{x \to 1^+} f(x)\), and \(\displaystyle \lim_{x \to 1} f(x)\).
\(x\) \(f(x)\) \(x\) \(f(x)\) 0.99 1.01 0.999 1.001 0.9999 1.0001
- Use a calculator to complete the table below for the function \(g(x) = (1+x)^{1/x}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 0^-} g(x)\), \(\displaystyle \lim_{x \to 0^+} g(x)\), and \(\displaystyle \lim_{x \to 0} g(x)\).
\(x\) \(g(x)\) \(x\) \(g(x)\) -0.01 0.01 -0.001 0.001 -0.0001 0.0001
- Use a calculator to complete the table below for the function \(h(x) = \dfrac{x+1}{x-2}\), then use your results to make inferences about \(\displaystyle \lim_{x \to 2^-} h(x)\), \(\displaystyle \lim_{x \to 2^+} h(x)\), and \(\displaystyle \lim_{x \to 2} h(x)\).
\(x\) \(h(x)\) \(x\) \(h(x)\) 1.99 2.01 1.999 2.001 1.9999 2.0001