Maths WorkshopMaths problems with full solutions

Limits

38 problems · Limits of functions, equivalents and expansions.

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Ex.22Three basic limits by conjugation and standard limits

#conjugate #standard limit #sine

Ex.51Four limits with absolute value and trigonometry

#absolute value #one-sided limits #factorization

Ex.52Three more classical limits

#squeeze theorem #logarithm #exponential form

Ex.69Three limits involving logarithm and exponentials

#exponential form #logarithm #indeterminate forms

Ex.192(e^(2x) − 1) over x

#exponential #change of variable

Ex.193A limit by factoring

#indeterminate forms #factoring

Ex.194Limits of rational functions at infinity

#limits at infinity #rational functions

Ex.195One-sided limits of |x|/x

#one-sided limits #absolute value

Ex.196sin(3x) over 5x

#fundamental trigonometric limit #change of variable

Ex.197Sine over x at infinity

#squeeze theorem #limits at infinity

Ex.314A limit with sine

#limits #taylor

Ex.318A limit with the exponential

#limits #taylor

Ex.344The conjugate trick at infinity

#conjugate #square roots #infinity minus infinity

Ex.345The conjugate trick at zero

#conjugate #square roots #indeterminate forms

Ex.346Epsilon-delta for a linear function

#epsilon-delta #definition of limit

Ex.347(e^x − cos x) over sin x

#combining known limits #indeterminate forms

Ex.348One minus cosine over x squared

#trigonometric limits #half-angle formula

Ex.351(2^x − 1) over x

#exponential #logarithm #derivative

Ex.352x ln(1 + 1/x) at infinity

#logarithm #number e #limits at infinity

Ex.353x times sin(1/x)

#squeeze theorem #oscillation

Ex.707Cosine to the power 1/x²

#exponential form #Taylor expansion #indeterminate forms

Ex.708Difference of two infinite reciprocals

#infinity minus infinity #Taylor expansion

Ex.709Epsilon-delta for x squared

#epsilon-delta #definition of limit #restricting delta

Ex.710The gap between two exponentials at infinity

#exponential #limits at infinity #change of variable

Ex.711The logarithm is slower than any power

#growth comparison #logarithm #inequalities

Ex.712(1 + sin x)^(1/x)

#exponential form #indeterminate forms #composition

Ex.713Polynomials versus the exponential

#growth comparison #exponential #squeeze theorem

Ex.714sin x − x cos x over x cubed

#Taylor expansion #little-o notation #cancellation

Ex.715sin(1/x) has no limit at 0

#sequences #non-existence of limits

Ex.716Tangent minus sine over x cubed

#trigonometric limits #factoring

Ex.717x ln x at zero, and x^x

#indeterminate forms #change of variable #logarithm

Ex.718x sin x has no limit at infinity

#sequences #non-existence of limits #infinite limits

Ex.719x times the Dirichlet function

#squeeze theorem #pathological functions #continuity

Ex.1464How fast does (1 + x)^(1/x) reach e?

#number e #Taylor expansion #order of convergence

Ex.1465Where power means lead

#power mean #geometric mean #logarithm #expansions

Ex.1466Reciprocal squares of x and tangent

#Taylor expansion #infinity minus infinity #tangent

Ex.1467The Dirichlet function has no limit anywhere

#sequences #density #non-existence of limits #pathological functions

Ex.2067Tangent of sine minus sine of tangent

#Taylor expansion #composition #high order #challenge