The limit of a function

First example and intuitive definition

Example: \(\lim_{x\to1}\frac{x-1}{x^2-1}\)

x f(x) x f(x)
0.9 0.526315789474 1.1 0.47619047619
0.999 0.500250125063 1.001 0.499750124938
0.99999 0.500002500013 1.00001 0.499997500012
0.9999999 0.50000002498 1.0000001 0.49999997502

The perils of subtraction

t f(t) t f(t)
-0.1 0.1666203960726864 0.1 0.1666203960726864
-0.001 0.1666666620547819 0.001 0.1666666620547819
-0.0000001 0.1665334536937735 0.0000001 0.1665334536937735
-0.00000000001 0 0.00000000001 0

\(\lim_{x\to0}\frac{\sin(x)}{x}\)

x f(x) x f(x)
-0.1 0.9983341664682815 0.1 0.9983341664682815
-0.001 0.9999998333333416 0.001 0.9999998333333416
-0.00001 0.9999999999833333 0.00001 0.9999999999833333
-0.0000001 0.9999999999999983 0.0000001 0.9999999999999983

\(\lim_{x\to0}\sin\frac{\pi}{x}\)

x f(x) x f(x)
-0.1 0.0000000000000001 0.1 -0.0000000000000001
-0.001 -0.0000000000000196 0.001 0.0000000000000196
-0.00001 0.0000000000012188 0.00001 -0.0000000000012188

One-sided limits

x f(x)
0.1 0.3162277660168379
0.001 0.0316227766016838
0.00001 0.0031622776601684

\(\lim_{x\to0}\frac{1}{x^2}\)

x f(x) x f(x)
-0.1 100 0.1 100
-0.001 1000000 0.001 1000000
-0.00001 10000000000 0.00001 10000000000

\(\lim_{x\to(\pi/2)^-}\tan(x)\) and \(\lim_{x\to(\pi/2)^+}\tan(x)\)

x f(x) x f(x)
\(\pi/2-0.1\) 9.9666444232592362 \(\pi/2+0.1\) -9.9666444232592379
\(\pi/2-0.001\) 999.999666666637836 \(\pi/2+0.001\) -999.9996666666493184
\(\pi/2-0.00001\) 99999.9999966482282616 \(\pi/2+0.00001\) -99999.9999967626499711