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));
|
|
|