2.2: The Limit of a Function

1. Sketch the graph of a function $f(x)$ that is continuous at $x = 1$ with $\displaystyle \lim_{x \to 1} f(x) = -2$.
    
2. Sketch the graph of a function $g(x)$ that is discontinuous at $x = 1$ with $\displaystyle \lim_{x \to 1} g(x) = -2$.
    
3. 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 $h(-2) = 3$.
    
4. Sketch the graph of a function $f(t)$ with $\displaystyle \lim_{t \to 0} f(t) = -\infty$.
    
5. 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 $g(1) = 2$.
    
6. 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.
    
7. 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.
    
8. 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.
    
9. 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.
    
10. 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.
     
11. 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.
     
12. 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

    
     
13. 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

    
     
14. 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

    
     
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 $\displaystyle \lim_{x \to 0^-} g(x)$, $\displaystyle \lim_{x \to 0^+} g(x)$, 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

    
     
16. 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