# Section 2.2

## 2536 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
 \newcommand{\Bold}{\mathbf{#1}}x \ {\mapsto}\ \frac{x - 2}{x^{2} - 4}
for x in [1.9, 1.99, 1.9999, 1.99999]: f(x);
 \newcommand{\Bold}{\mathbf{#1}}0.256410256410257 \newcommand{\Bold}{\mathbf{#1}}0.250626566416041 \newcommand{\Bold}{\mathbf{#1}}0.250006250156216 \newcommand{\Bold}{\mathbf{#1}}0.250000625001614
for x in [2.1, 2.01, 2.001, 2.0001]: f(x);
 \newcommand{\Bold}{\mathbf{#1}}0.243902439024390 \newcommand{\Bold}{\mathbf{#1}}0.249376558603493 \newcommand{\Bold}{\mathbf{#1}}0.249937515621086 \newcommand{\Bold}{\mathbf{#1}}0.249993750156284

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
 \newcommand{\Bold}{\mathbf{#1}}x \ {\mapsto}\ \frac{\sin\left(x\right)}{x}
plot(f,(x,-1,1)); for x in [0.1, 0.01, 0.001, 0.0001, 0.000000000000000000000000000000000001]: f(x);
 \newcommand{\Bold}{\mathbf{#1}}0.998334166468282 \newcommand{\Bold}{\mathbf{#1}}0.999983333416666 \newcommand{\Bold}{\mathbf{#1}}0.999999833333342 \newcommand{\Bold}{\mathbf{#1}}0.999999998333333 \newcommand{\Bold}{\mathbf{#1}}1.00000000000000
for x in [-0.1, -0.01, -0.001, -0.0001, -0.000000000000000000000000000000000001]: f(x);
 \newcommand{\Bold}{\mathbf{#1}}0.998334166468282 \newcommand{\Bold}{\mathbf{#1}}0.999983333416666 \newcommand{\Bold}{\mathbf{#1}}0.999999833333342 \newcommand{\Bold}{\mathbf{#1}}0.999999998333333 \newcommand{\Bold}{\mathbf{#1}}1.00000000000000

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
 \newcommand{\Bold}{\mathbf{#1}}x \ {\mapsto}\ \sin\left(\frac{1}{x}\right)
for x in [pi, pi/2, pi/4, pi/6, pi/8, pi/10, pi/100, pi/10000000]: RR(f(x));
 \newcommand{\Bold}{\mathbf{#1}}0.312961796207787 \newcommand{\Bold}{\mathbf{#1}}0.594480768524822 \newcommand{\Bold}{\mathbf{#1}}0.956055657327630 \newcommand{\Bold}{\mathbf{#1}}0.943066732256947 \newcommand{\Bold}{\mathbf{#1}}0.560602798839214 \newcommand{\Bold}{\mathbf{#1}}-0.0414942916985869 \newcommand{\Bold}{\mathbf{#1}}0.403246742300302 \newcommand{\Bold}{\mathbf{#1}}-0.491569491718270
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
 \newcommand{\Bold}{\mathbf{#1}}x \ {\mapsto}\ \frac{1}{{\left(x - 2\right)}^{2}}
plot(f,(x,1,1.9))+ plot(f,(x,2.1,3)); 