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Mathematics LibreTexts

2.2: The Limit of a Function

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  1. Sketch the graph of a function f(x) that is continuous at x=1 with limx1f(x)=2.
     
  2. Sketch the graph of a function g(x) that is discontinuous at x=1 with limx1g(x)=2.
     
  3. Sketch the graph of a function h(x) with limx2h(x)=3, limx2+h(x)=1, and h(2)=3.
     
  4. Sketch the graph of a function f(t) with limt0f(t)=.
     
  5. Sketch the graph of a function g(t) with limt1g(t)=, limt1+g(t)=0, and g(1)=2.
     
  6. Given only that limx11h(x)=4 for some function h(x), is it possible to calculate limx11h(x) and limx11+h(x)? Briefly explain why or why not.
     
  7. Given only that limx11f(x)=4 for some function f(x), is it possible to calculate f(11)? Briefly explain why or why not.
     
  8. Given only that g(11)=4 for some function g(x), is it possible to calculate limx11g(x)? Briefly explain why or why not.
     
  9. Given only that h(11)=4 for some continuous function h(x), is it possible to calculate limx11h(x)? Briefly explain why or why not.
     
  10. Given only that limx11f(x)=4 and limx11+f(x)=4 for some function f(x), is it possible to calculate limx11f(x)? Briefly explain why or why not.
     
  11. Given only that limx11f(x)=5 and limx11+f(x)=2 for some function f(x), is it possible to calculate limx11f(x)? Briefly explain why or why not.
     
  12. Use a calculator to complete the table below for the function g(x)=2x+1, then use your results to make inferences about limx1g(x), limx1+g(x), and limx1g(x).
     
    x g(x)   x g(x)
    -1.01     -0.99  
    -1.001     -0.999  
    -1.0001     -0.9999  

     
  13. Use a calculator to complete the table below for the function h(x)=sin(x)x, then use your results to make inferences about limx0h(x), limx0+h(x), and limx0h(x).
     
    x h(x)   x h(x)
    -0.01     0.01  
    -0.001     0.001  
    -0.0001     0.0001  

     
  14. Use a calculator to complete the table below for the function f(x)=x21)|x1|, then use your results to make inferences about limx1f(x), limx1+f(x), and limx1f(x).
     
    x f(x)   x f(x)
    0.99     1.01  
    0.999     1.001  
    0.9999     1.0001  

     
  15. 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 limx0g(x), limx0+g(x), and limx0g(x).
     
    x g(x)   x g(x)
    -0.01     0.01  
    -0.001     0.001  
    -0.0001     0.0001  

     
  16. Use a calculator to complete the table below for the function h(x)=x+1x2, then use your results to make inferences about limx2h(x), limx2+h(x), and limx2h(x).
     
    x h(x)   x h(x)
    1.99     2.01  
    1.999     2.001  
    1.9999     2.0001  

2.2: The Limit of a Function is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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