Number and Algebra
Functions
Geometry and Trigonometry
Statistics and Probability
Calculus
Evaluate the limits limx→0+1xandlimx→0−1x.\lim_{x\to0^+}\frac{1}{x}\quad\text{and}\quad\lim_{x\to0^-}\frac{1}{x}.limx→0+x1andlimx→0−x1.
Determine limx→1−∣x−1∣x−1andlimx→1+∣x−1∣x−1.\lim_{x\to1^-}\frac{|x-1|}{x-1}\quad\text{and}\quad\lim_{x\to1^+}\frac{|x-1|}{x-1}. limx→1−x−1∣x−1∣andlimx→1+x−1∣x−1∣.
Find limx→3−1(x−3)2andlimx→3+1(x−3)2.\lim_{x\to3^-}\frac{1}{(x-3)^2}\quad\text{and}\quad\lim_{x\to3^+}\frac{1}{(x-3)^2}. limx→3−(x−3)21andlimx→3+(x−3)21.
Evaluate one-sided limits: limx→2−x−2∣x−2∣andlimx→2+x−2∣x−2∣.\lim_{x\to2^-}\frac{x-2}{|x-2|}\quad\text{and}\quad\lim_{x\to2^+}\frac{x-2}{|x-2|}. limx→2−∣x−2∣x−2andlimx→2+∣x−2∣x−2.
Evaluate the one-sided limits limx→2−x2−4x−2andlimx→2+x2−4x−2.\lim_{x\to2^-}\frac{x^2-4}{x-2}\quad\text{and}\quad\lim_{x\to2^+}\frac{x^2-4}{x-2}. limx→2−x−2x2−4andlimx→2+x−2x2−4.
Determine the one-sided limits of the natural logarithm: limx→0+ln(x)andlimx→0−ln(x).\lim_{x\to0^+}\ln(x)\quad\text{and}\quad\lim_{x\to0^-}\ln(x).limx→0+ln(x)andlimx→0−ln(x).
Consider the piecewise function f(x)={x2,x<1,2x−1,x≥1.f(x)=\begin{cases}x^2,&x<1,\\2x-1,&x\ge1.\end{cases}f(x)={x2,2x−1,x<1,x≥1. Compute limx→1−f(x)andlimx→1+f(x).\lim_{x\to1^-}f(x)\quad\text{and}\quad\lim_{x\to1^+}f(x).limx→1−f(x)andlimx→1+f(x).
Evaluate the one-sided limits of limx→0+xsin(1/x)andlimx→0−xsin(1/x).\lim_{x\to0^+}x\sin\bigl(1/x\bigr)\quad\text{and}\quad\lim_{x\to0^-}x\sin\bigl(1/x\bigr).limx→0+xsin(1/x)andlimx→0−xsin(1/x).
Evaluate the two-sided behaviour of limx→0+e1/xandlimx→0−e1/x.\lim_{x\to0^+}e^{1/x}\quad\text{and}\quad\lim_{x\to0^-}e^{1/x}. limx→0+e1/xandlimx→0−e1/x.
Evaluate the limits limx→π−tan(x)andlimx→π+tan(x).\lim_{x\to\pi^-}\tan(x)\quad\text{and}\quad\lim_{x\to\pi^+}\tan(x).limx→π−tan(x)andlimx→π+tan(x).
Find the behaviour of the oscillating function limx→0−sin(1x)andlimx→0+sin(1/x).\lim_{x\to0^-}\sin\bigl(\tfrac1x\bigr)\quad\text{and}\quad\lim_{x\to0^+}\sin\bigl(1/x\bigr).limx→0−sin(x1)andlimx→0+sin(1/x).
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