Limits of Functions

Given the polynomials:

\$\$p(x) = a_kx^k + a_{k-1}x^{k-1} + ... + a_1x + a_0\$\$ \$\$q(x) = b_mx^m + b_{m-1}x^{m-1} + ... + b_1x + b_0\$\$

then:

\$\${lim}↙{x→x_0} p(x) = p(x_0) \$\$ \$\${lim}↙{x→x_0} {p(x)}/{q(x)} = {p(x_0)}/{q(x_0)}\$\$

if \$q(x_0) ≠ 0\$\$ \$\${lim}↙{x→±∞} {p(x)}/{q(x)} = [\table ± ∞, k > m; {a_k}/{b_m}, k = m; 0, k = m, k < m;\$\$

e.g. \${lim}↙{x→± ∞} {3x^2}/{x^2 + 5}\$

In this case, k = 2 and m = 2, so \${lim}↙{x→± ∞} {3x^2}/{x^2 + 5} = {a_k}/{b_m} = {3}/{1} = 3\$

e.g. \${lim}↙{x→5} {-x^2 + 5x}/{x^2 - 2x - 15} = {-x(x - 5)}/{(x - 5)(x + 3)} = {-x}/{(x + 3)} = -5/8\$

 \${lim}↙{x→∞} {x^3 - 1}/{x^2 + x}\$ and, \${lim}↙{x→-∞} {x^3 - 1}/{x^2 + x}\$ In this case, k = 3 and m = 2, so \${lim}↙{x→∞} {x^3 - 1}/{x^2 + x} = ∞\$ \${lim}↙{x→-∞} {x^3 - 1}/{x^2 + x} = -∞\$

some important limits

\${lim}↙{x→0} {sinx}/x = 1\$

\${lim}↙{x→± ∞} (1 + 1/x)^x = e\$

\${lim}↙{x→0} {log_a(1 + x)}/x = log_ae\$, provided \$0 < a ≠ 1\$

\${lim}↙{x→0} {ln(1 + x)}/x = 1\$

\${lim}↙{x→0} {a^x - 1}/x = ln (a)\$, provided \$0 < a ≠ 1\$

\${lim}↙{x→0} {e^x - 1}/x = 1\$

Asymptotes

If \${lim}↙{x→∞} (f(x)-g(x)) = 0\$, g approximates f asymptotically when \$x→∞\$

\$m = {lim}↙{x→∞} {f(x)}/x\$

\$q = {lim}↙{x→∞} (f(x) - m⋅x)\$

Example

f: x|→ \${x^3 - 2x^2 - x + 3}/{x^2 - 1}\$, \$D_f\$ = R\{±1}

Vertical asymptotes: x = ± 1

Oblique asymptote:

\$m = {lim}↙{x→±∞} {f(x)}/x = {x^3 - 2x^2 - x + 3}/{x^3 - x} = 1\$

\$q = {lim}↙{x→±∞} (f(x) - m⋅x)\$

\$= {lim}↙{x→±∞} {{x^3 - 2x^2 - x + 3 - x^3 + x}/{x^2 - 1} = -2\$

Therefore, y = x - 2 is the oblique asymptote.

Example

g: x|→ \${x^5 - x + 1}/{x^3 + x}\$, \$D_g = R^+\$

Vertical asymptote: x = 0

Carrying through the division:

\$g(x) = x^2 - 1 + 1/{x^3 + x}\$

For which \${lim}↙{x→±∞} (g(x) - x^2 + 1)\$

\$ = {lim}↙{x→±∞} 1/{x^3 + x} = 0\$

The function is therefore approximately asymptotic to the parabola \$y = x^2 - 1\$

Site Index

Latest Item on Science Library:

The most recent article is:

Trigonometry

View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

Environment

Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on ScienceLibrary.info.

Great Scientists

Rebecca Harms

b. 1956

Rebecca Harms, born 1956 in Lower Saxony, is a German environmentalist and Member of the European Parliament.

Quote of the day...

The paleoclimate record shouts out to us that, far from being self-stabilizing, the Earth's climate system is an ornery beast which overreacts even to small nudges.