Common limits, with the method

A solver returns the number. Each page here also names WHICH technique the limit wants and why the obvious first move fails, because choosing the method is the part students actually get stuck on.

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$the special trigonometric limit
• $\lim_{x \to 0} \frac{1-\cos x}{x} = 0$the special trigonometric limit
• $\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}$L'Hopital's rule applied twice
• $\lim_{x \to 0} \frac{\tan x}{x} = 1$rewriting as sine over cosine
• $\lim_{x \to 0} \frac{\sin 2x}{x} = 2$matching the special trigonometric limit
• $\lim_{x \to 0} \frac{\sin 3x}{\sin 5x} = \frac{3}{5}$the special trigonometric limit applied twice
• $\lim_{x \to 0} \frac{\tan x}{\sin x} = 1$algebraic simplification
• $\lim_{x \to 0} \frac{e^x-1}{x} = 1$the limit definition of the derivative
• $\lim_{x \to 1} \frac{x^2-1}{x-1} = 2$factoring and cancelling
• $\lim_{x \to 2} \frac{x^3-8}{x-2} = 12$factoring the difference of cubes
• $\lim_{x \to 2} \frac{x^2-4}{x^2-x-2} = \frac{4}{3}$factoring both polynomials
• $\lim_{x \to 4} \frac{\sqrt{x}-2}{x-4} = \frac{1}{4}$multiplying by the conjugate
• $\lim_{x \to \infty} \left(1+\frac{1}{x}\right)^x = e$the definition of e
• $\lim_{x \to 0^+} (1+x)^{1/x} = e$the definition of e
• $\lim_{x \to 0^+} x^x = 1$logarithms then L'Hopital
• $\lim_{x \to 0^+} x\ln x = 0$rewriting as a quotient then L'Hopital
• $\lim_{x \to 0^+} \ln x = -\infty$end behaviour of the logarithm
• $\lim_{x \to 0} \frac{1}{x^2} = \infty$unbounded behaviour at a vertical asymptote
• $\lim_{x \to \infty} \frac{1}{x} = 0$end behaviour
• $\lim_{x \to \infty} \frac{2x^2+3}{x^2-1} = 2$comparing leading degrees
• $\lim_{x \to \infty} \frac{3x+1}{x^2+2} = 0$comparing leading degrees
• $\lim_{x \to \infty} \frac{x^2}{x+1} = \infty$comparing leading degrees
• $\lim_{x \to -\infty} e^x = 0$end behaviour of the exponential
• $\lim_{x \to \infty} \frac{\ln x}{x} = 0$L'Hopital's rule
• $\lim_{x \to \infty} \frac{\sin x}{x} = 0$the squeeze theorem
• $\lim_{x \to \infty} \arctan x = \frac{\pi}{2}$the range of the arctangent
• $\lim_{x \to \infty} \frac{x}{\sqrt{x^2+1}} = 1$dividing by the dominant term
• $\lim_{x \to \infty} \sqrt{x^2+x}-x = \frac{1}{2}$multiplying by the conjugate
• $\lim_{x \to 0} x\sin\frac{1}{x} = 0$the squeeze theorem
• $\lim_{x \to 0} x^2\sin\frac{1}{x} = 0$the squeeze theorem
• $\lim_{x \to 0} \frac{x-\sin x}{x^3} = \frac{1}{6}$L'Hopital's rule applied three times
• $\lim_{x \to 0} \frac{\tan x-x}{x^3} = \frac{1}{3}$L'Hopital's rule applied three times
• $\lim_{x \to 0} \frac{1-\cos 2x}{x^2} = 2$the special trigonometric limit
• $\lim_{x \to 0} \frac{\sqrt{x+1}-1}{x} = \frac{1}{2}$multiplying by the conjugate
• $\lim_{x \to 0} \frac{\sin x}{x+x^2} = 1$the special trigonometric limit
• $\lim_{x \to 0} \frac{\ln(x+1)}{x} = 1$the limit definition of the derivative
• $\lim_{x \to 0} \frac{2^x-1}{x} = \ln 2$the limit definition of the derivative
• $\lim_{x \to 0} \frac{e^{2x}-1}{x} = 2$the limit definition of the derivative
• $\lim_{x \to 3} \frac{x-3}{x^2-9} = \frac{1}{6}$factoring and cancelling
• $\lim_{x \to 2} \frac{\frac{1}{x}-\frac{1}{2}}{x-2} = -\frac{1}{4}$clearing the complex fraction
• $\lim_{x \to \infty} \frac{x}{e^x} = 0$L'Hopital's rule
• $\lim_{x \to \infty} x^2e^{-x} = 0$L'Hopital's rule applied twice
• $\lim_{x \to \infty} \frac{(\ln x)^2}{x} = 0$L'Hopital's rule applied twice
• $\lim_{x \to \infty} \frac{5x^3-2x}{3x^3+7} = \frac{5}{3}$comparing leading degrees
• $\lim_{x \to \infty} \frac{x^2+1}{2x^2-3} = \frac{1}{2}$comparing leading degrees
• $\lim_{x \to \infty} \frac{2x+3}{\sqrt{x^2+1}} = 2$dividing by the dominant term
• $\lim_{x \to \infty} \frac{3^x}{4^x} = 0$combining into a single base
• $\lim_{x \to \infty} x^{1/x} = 1$logarithms then L'Hopital
• $\lim_{x \to \infty} \frac{\sqrt{x^2+1}}{x} = 1$dividing by the dominant term
• $\lim_{x \to 0^+} \frac{1}{x} = \infty$one-sided unbounded behaviour
• $\lim_{x \to 0^-} \frac{1}{x} = -\infty$one-sided unbounded behaviour
• $\lim_{x \to 0^+} \frac{|x|}{x} = 1$evaluating the piecewise definition
• $\lim_{x \to 0^-} \frac{|x|}{x} = -1$evaluating the piecewise definition
• $\lim_{x \to \infty} \ln x = \infty$end behaviour of the logarithm
• $\lim_{x \to -\infty} \arctan x = -\frac{\pi}{2}$the range of the arctangent
• $\lim_{x \to -\infty} x^3 = -\infty$end behaviour of an odd power
• $\lim_{x \to \infty} \frac{\cos x}{x} = 0$the squeeze theorem
• $\lim_{x \to 0} \frac{x}{\sin x} = 1$the reciprocal of the special trigonometric limit
• $\lim_{x \to 0} \frac{\sin 5x}{3x} = \frac{5}{3}$matching the inner angle to the denominator
• $\lim_{x \to 0} \frac{\arcsin x}{x} = 1$substituting the angle, then the special trigonometric limit
• $\lim_{x \to 0} \frac{\arctan x}{x} = 1$L'Hopital's rule, or the Maclaurin series
• $\lim_{x \to 0} \frac{\tan 3x}{x} = 3$splitting tangent into sine over cosine
• $\lim_{x \to 0} \frac{\sin\left(x^{2}\right)}{x} = 0$matching the inner angle, leaving a stray factor
• $\lim_{x \to 0} \frac{\sin x}{x^{3}} = \infty$peeling off the special trigonometric limit
• $\lim_{x \to 0} \frac{\sec x - 1}{x} = 0$recognising it as a derivative at a point
• $\lim_{x \to 1} \frac{\ln x}{x - 1} = 1$recognising it as a derivative at a point
• $\lim_{x \to 0} \frac{e^{x} - e^{-x}}{x} = 2$splitting into two standard exponential limits
• $\lim_{x \to 0} \frac{\cos x - 1}{\sin x} = 0$dividing through by x to expose the standard limits
• $\lim_{x \to 0} \frac{\ln(1 + 2x)}{x} = 2$matching the inner expression to the denominator
• $\lim_{x \to 1} \frac{x - 1}{\sqrt{x} - 1} = 2$factoring a difference of squares
• $\lim_{x \to 0} \frac{\sqrt{x+4} - 2}{x} = \frac{1}{4}$multiplying by the conjugate
• $\lim_{x \to 2} \frac{x - 2}{\sqrt{x+2} - 2} = 4$multiplying by the conjugate of the denominator
• $\lim_{x \to \infty} \frac{x}{x+1} = 1$dividing by the highest power
• $\lim_{x \to \infty} \frac{3x^{2} + 2x}{x^{2} - 5} = 3$dividing by the highest power
• $\lim_{x \to \infty} \frac{x + \sin x}{x} = 1$splitting the fraction, then the squeeze theorem
• $\lim_{x \to \infty} \sqrt{x^{2}-x} - x = -\frac{1}{2}$multiplying by the conjugate
• $\lim_{x \to 0^+} \cot x = \infty$one-sided analysis of cosine over sine
• $\lim_{x \to \infty} e^{-x} = 0$rewriting as a reciprocal
• $\lim_{x \to -\infty} 5^{x} = 0$rewriting as a reciprocal
• $\lim_{x \to \infty} \sqrt{x} = \infty$unbounded growth of a power function
• $\lim_{x \to 0} \frac{\cos x - 1}{x^{2}} = -\frac{1}{2}$L'Hopital's rule twice, or the Maclaurin series
• $\lim_{x \to 0} \frac{e^{x}-1-x}{x^{2}} = \frac{1}{2}$L'Hopital's rule twice, or the Maclaurin series
• $\lim_{x \to 0} \frac{\sin x - x}{x} = 0$splitting the fraction into a known limit
• $\lim_{x \to \infty} \frac{x^{2}}{e^{x}} = 0$L'Hopital's rule twice
• $\lim_{x \to \infty} \frac{x^{3}}{e^{x}} = 0$L'Hopital's rule three times
• $\lim_{x \to \infty} \sqrt{x+1}-\sqrt{x} = 0$multiplying by the conjugate
• $\lim_{x \to \infty} x\ln\left(1+\frac{1}{x}\right) = 1$substitution to a standard logarithm limit
• $\lim_{x \to \infty} \left(1-\frac{1}{x}\right)^{x} = \frac{1}{e}$the compound interest limit with k = -1
• $\lim_{x \to \infty} \left(1+\frac{2}{x}\right)^{x} = e^{2}$the compound interest limit with k = 2
• $\lim_{x \to 0} \frac{\tan x}{x^{3}} = \infty$peeling off the standard trigonometric limit
• $\lim_{x \to \infty} \frac{\arctan x}{\sqrt{x}} = 0$bounded over unbounded
• $\lim_{x \to \infty} \frac{\ln(x+1)}{\ln x} = 1$L'Hopital's rule
• $\lim_{x \to \infty} \sqrt{x^{2}+4}-x = 0$multiplying by the conjugate
• $\lim_{x \to \infty} \frac{x}{\ln x} = \infty$L'Hopital's rule
• $\lim_{x \to 0} \frac{\sin 3x}{\tan 2x} = \frac{3}{2}$matching each inner angle to the standard limits
• $\lim_{x \to 0} \frac{1-\cos x}{x\sin x} = \frac{1}{2}$dividing through by x squared
• $\lim_{x \to 0} \frac{\sin x}{\tan x} = 1$simplifying to cos x, then direct substitution
• $\lim_{x \to 9} \frac{\sqrt{x}-3}{x-9} = \frac{1}{6}$factoring a difference of squares
• $\lim_{x \to 1} \frac{e^{x}-e}{x-1} = e$recognising it as a derivative at a point
• $\lim_{x \to 0^+} \frac{\ln(\sin x)}{\ln x} = 1$L'Hopital's rule, or the leading-order comparison
• $\lim_{x \to 3} \frac{x^{2}-9}{x-3} = 6$factoring the difference of squares
• $\lim_{x \to -1} \frac{x^{3}+1}{x+1} = 3$factoring the sum of cubes
• $\lim_{x \to 2} \frac{x^{4}-16}{x-2} = 32$factoring twice, or recognising a derivative at a point
• $\lim_{x \to 0} \frac{\sqrt{x+9}-3}{x} = \frac{1}{6}$multiplying by the conjugate
• $\lim_{x \to 4} \frac{x-4}{\sqrt{x}-2} = 4$factoring the numerator as a difference of squares in the root
• $\lim_{x \to 1} \frac{\sin\left(\pi x\right)}{x-1} = -\pi$substituting u = x - 1, then the special trigonometric limit
• $\lim_{x \to 0} \frac{e^{x}-1}{\sin x} = 1$splitting into two standard limits
• $\lim_{x \to 0} \frac{\ln\left(\cos x\right)}{x^{2}} = -\frac{1}{2}$L'Hopital's rule applied twice
• $\lim_{x \to \infty} \sin\left(\frac{1}{x}\right) = 0$continuity of sine at 0
• $\lim_{x \to 0^+} x^{\sin x} = 1$taking logarithms, then rewriting as a quotient for L Hopital
• $\lim_{x \to \infty} \frac{\sqrt{4x^{2}+1}}{x+3} = 2$dividing by the dominant term
• $\lim_{x \to \infty} \frac{3x-2}{\sqrt{2x^{2}+1}} = \frac{3\sqrt{2}}{2}$dividing by the dominant term
• $\lim_{x \to 1^+} \frac{1}{x-1} = \infty$one-sided analysis at a vertical asymptote
• $\lim_{x \to 3} \frac{x^{2}-x-6}{x-3} = 5$factoring the quadratic
• $\lim_{x \to \infty} \sqrt{x^{2}+3x}-x = \frac{3}{2}$multiplying by the conjugate
• $\lim_{x \to 0^+} \frac{\sin x}{\sqrt{x}} = 0$splitting off the special trigonometric limit
• $\lim_{x \to 0} \frac{\tan 2x}{\tan 3x} = \frac{2}{3}$matching each inner angle to the standard limits
• $\lim_{x \to 0} \frac{\arcsin 2x}{x} = 2$matching the inner angle to the denominator
• $\lim_{x \to 0} \frac{\arctan 3x}{5x} = \frac{3}{5}$matching the inner angle to the denominator
• $\lim_{x \to 0} \frac{e^{3x}-1}{x} = 3$matching the exponent to the denominator
• $\lim_{x \to 0} \frac{\ln\left(1+x^{2}\right)}{x^{2}} = 1$substituting u = x squared into the standard logarithm limit
• $\lim_{x \to 0} \frac{x^{2}}{1-\cos x} = 2$the reciprocal of the standard cosine limit
• $\lim_{x \to \infty} x^{2}\sin\left(\frac{1}{x}\right) = \infty$peeling off the standard limit, leaving one factor of x
• $\lim_{x \to \infty} \frac{\sqrt{x+1}}{\sqrt{x}} = 1$combining under a single root
• $\lim_{x \to \frac{\pi}{2}^-} \tan x = \infty$one-sided analysis of sine over cosine
• $\lim_{x \to \frac{\pi}{2}^-} \sec x = \infty$reciprocal of a vanishing positive quantity
• $\lim_{x \to 0^+} \csc x = \infty$reciprocal of a vanishing positive quantity
• $\lim_{x \to 1^+} \frac{x}{x-1} = \infty$one-sided analysis at a vertical asymptote
• $\lim_{x \to 2^+} \frac{1}{x^{2}-4} = \infty$factoring to isolate the vanishing factor
• $\lim_{x \to 0^-} e^{1/x} = 0$tracking the exponent, then end behaviour of the exponential
• $\lim_{x \to 0^+} \arctan\!\left(\frac{1}{x}\right) = \frac{\pi}{2}$tracking the inner expression, then the range of the arctangent
• $\lim_{x \to 0} \frac{1-\cos 3x}{x^{2}} = \frac{9}{2}$matching the inner angle to the denominator
• $\lim_{x \to 0} \frac{\sin 4x}{\sin 2x} = 2$matching each inner angle to the standard limits
• $\lim_{x \to 0} \frac{\tan 5x}{x} = 5$matching the inner angle, leaving a stray factor
• $\lim_{x \to 0} \frac{\arcsin 3x}{x} = 3$matching the inner expression to the denominator
• $\lim_{x \to 0} \frac{\sin x}{x + \sin x} = \frac{1}{2}$dividing through by x to expose the standard limits
• $\lim_{x \to 0} \frac{x}{\tan x} = 1$the reciprocal of the standard tangent limit
• $\lim_{x \to 0} \frac{\sin^{2} x}{x} = 0$splitting off the special trigonometric limit
• $\lim_{x \to 0} \frac{\sin^{2} x}{x^{2}} = 1$the square of the special trigonometric limit
• $\lim_{x \to 0} \frac{3^{x}-1}{x} = \ln 3$the limit definition of the derivative
• $\lim_{x \to 0} \frac{\ln(1+3x)}{x} = 3$substitution to a standard logarithm limit
• $\lim_{x \to \infty} e^{-x^{2}} = 0$end behaviour of the exponential
• $\lim_{x \to \infty} x^{1/\ln x} = e$taking logarithms, which collapses the exponent exactly
• $\lim_{x \to \infty} \frac{x^{3}}{x^{2}+1} = \infty$comparing leading degrees
• $\lim_{x \to 4} \frac{x^{2}-16}{x-4} = 8$factoring a difference of squares
• $\lim_{x \to 3} \frac{x^{3}-27}{x-3} = 27$factoring the difference of cubes
• $\lim_{x \to 1} \frac{\sqrt{x+3}-2}{x-1} = \frac{1}{4}$multiplying by the conjugate
• $\lim_{x \to \frac{1}{2}} \frac{4x^{2}-1}{2x-1} = 2$factoring a difference of squares
• $\lim_{x \to \infty} \frac{\sqrt{9x^{2}+1}}{x} = 3$dividing by the dominant term
• $\lim_{x \to 0} x\cos\!\left(\frac{1}{x}\right) = 0$the squeeze theorem
• $\lim_{x \to \infty} \frac{\sin x}{x^{2}} = 0$bounded over unbounded
• $\lim_{x \to \pi} \frac{\tan x}{x-\pi} = 1$shifting with the period, then the standard limit