Section 2.2

1984 days ago by jimlb

What is the limit of $f(x) = x^2$ as $x \rightarrow 2$?

 
       
f(x) = x^2 
       
for x in [1.9, 1.99, 1.999, 1.99999, 1.9999999999999999999999]: f(x); 
       
3.61000000000000
3.96010000000000
3.99600100000000
3.99996000010000
4.000000000000000000000
3.61000000000000
3.96010000000000
3.99600100000000
3.99996000010000
4.000000000000000000000
for x in [2.1, 2.01, 2.001, 2.0001, 2.0000000000001]: f(x); 
       
4.41000000000000
4.04010000000000
4.00400100000000
4.00040001000000
4.00000000000040
4.41000000000000
4.04010000000000
4.00400100000000
4.00040001000000
4.00000000000040

From this it looks like the limit is 4.

What is the limit of $f(x) = \frac{x-2}{x^2 - 4}$ as $x \rightarrow 2$?

f(x) = (x-2)/(x^2-4);f 
       
for x in [1.9, 1.99, 1.9999, 1.99999]: f(x); 
       



for x in [2.1, 2.01, 2.001, 2.0001]: f(x); 
       



From this it looks like the limit is $0.25$.

Consider the function $f(x) = \frac{\sin x}{x}$.  What is the limit of this as $x \rightarrow 0$?

f(x) = sin(x)/x;f 
       
plot(f,(x,-1,1)); 
       
for x in [0.1, 0.01, 0.001, 0.0001, 0.000000000000000000000000000000000001]: f(x); 
       




for x in [-0.1, -0.01, -0.001, -0.0001, -0.000000000000000000000000000000000001]: f(x); 
       




This looks like the limit should be 1.

Consider the function $f(x) = \sin(1/x)$.  What is the limit as $x \rightarrow 0$?

f(x) = sin(1/x);f 
       
for x in [pi, pi/2, pi/4, pi/6, pi/8, pi/10, pi/100, pi/10000000]: RR(f(x)); 
       







plot(f, (x, -.0025, .0025)); 
       

Since this does not approach any number (it bounces up and down), the limit does not exist!  (We write this as DNE.)

f(x) = 1/(x-2)^2;f 
       
plot(f,(x,1,1.9))+ plot(f,(x,2.1,3));