Special Limits

Scientific Notebook also handles one-sided limits , limits at infinity, and infinite limits.

$\blacktriangleright$ Evaluate

$\lim\limits_{{x\rightarrow \infty }% }^{}$${\dfrac{{\sqrt{x^{2}+3x+5}}}{{\sqrt{16x^{2}+5x-3}}}}$ = ${\dfrac{{1}}{{4}}}$

$\lim\limits_{{x\rightarrow 0^{-}}}^{}$${\dfrac{{x}}{{\left\vert x\right\vert }}}$ = - 1

$\lim\limits_{{x\rightarrow 0^{+}}}^{}$${\dfrac{{x}}{{\left\vert x\right\vert }}}$ = 1

$\lim\limits_{{x\rightarrow 2}}^{}$${\dfrac{{x+2}}{{x-2}}}$ = $\func$undefined

$\lim\limits_{{x\rightarrow 2^{+}}}^{}$${\dfrac{{x+2}}{{x-2}}}$ = ∞

$\lim\limits_{{x\rightarrow 2^{-}}}^{}$${\dfrac{{x+2}}{{x-2}}}$ = - ∞

$\lim\limits_{{x\rightarrow 0}}^{}$sin$\left(\vphantom{ \dfrac{1}{x}}\right.$${\dfrac{{1}}{{x}}}$$\left.\vphantom{ \dfrac{1}{x}}\right)$ = -1..1


The notation -1..1 used in the last limit indicates the interval -1 ≤ x ≤ 1. Although this output is somewhat unusual, it is explained by the graph of y = sin${\frac{{1}}{{x}}}$. As x approaches 0, the function values of sin${\frac{{1}}{{x}}}$ include every number in the interval -1 ≤ x ≤ 1 infinitely often.

$\blacktriangleright$ Plot 2D + Rectangular

sin${\frac{{1}}{{x}}}$

dtbpFU3in2.0003in0pt y = sin${\frac{{1}}{{x}}}$