2.2: The Limit of a Function
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- Sketch the graph of a function f(x) that is continuous at x=1 with limx→1f(x)=−2.
- Sketch the graph of a function g(x) that is discontinuous at x=1 with limx→1g(x)=−2.
- Sketch the graph of a function h(x) with limx→−2−h(x)=3, limx→−2+h(x)=1, and h(−2)=3.
- Sketch the graph of a function f(t) with limt→0f(t)=−∞.
- Sketch the graph of a function g(t) with limt→−1−g(t)=∞, limt→−1+g(t)=0, and g(1)=2.
- Given only that limx→11h(x)=4 for some function h(x), is it possible to calculate limx→11−h(x) and limx→11+h(x)? Briefly explain why or why not.
- Given only that limx→11f(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 limx→11g(x)? Briefly explain why or why not.
- Given only that h(11)=4 for some continuous function h(x), is it possible to calculate limx→11h(x)? Briefly explain why or why not.
- Given only that limx→11−f(x)=4 and limx→11+f(x)=4 for some function f(x), is it possible to calculate limx→11f(x)? Briefly explain why or why not.
- Given only that limx→11−f(x)=5 and limx→11+f(x)=2 for some function f(x), is it possible to calculate limx→11f(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 limx→−1−g(x), limx→−1+g(x), and limx→−1g(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)=sin(x)x, then use your results to make inferences about limx→0−h(x), limx→0+h(x), and limx→0h(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)=x2−1)|x−1|, then use your results to make inferences about limx→1−f(x), limx→1+f(x), and limx→1f(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 limx→0−g(x), limx→0+g(x), and limx→0g(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)=x+1x−2, then use your results to make inferences about limx→2−h(x), limx→2+h(x), and limx→2h(x).
x h(x) x h(x) 1.99 2.01 1.999 2.001 1.9999 2.0001