Limit of Trigonometric / Logarithmic / Exponential Functions

Limit of Trigonometric Functions

» limx→0sinxx=1limx→0sinxx=1

» limx→0arcsinxx=1limx→0arcsinxx=1

» limx→01-cosxx=0limx→01−cosxx=0

Limit of Logarithmic Functions

» limx→0ln(1+x)x=1limx→0ln(1+x)x=1

» limx→0ax-1x=ln(a)limx→0ax−1x=ln(a)

Limit of Exponential Functions

» limx→∞(1+ax)x=ealimx→∞(1+ax)x=ea

» limx→0ex-1x=1

» limx→∞ax=(∞ifa>1), and (0ifa<1)

» limx→0(1+x)n-1x=n

» limx→0sin-1xx=1

problem

How to find the expected value for f(x)=sinxx at x=0?

When a function evaluates to 00 at an input value, the common factors of the numerator and denominators are canceled to calculate the limit of the function at the input value. That works only if the numerator and denominator are polynomials.

When one of the numerator or denominator is a trigonometric function, how to compute the limits?

multiple solutions

There are multiple proofs for limx→0sinxx=1.

• Substitute series expansion

sinx=x-x33!+x55!+⋯

• Geometrically prove that

cosx<sinxx<1

• Use the L'hospital's rule to differentiate numerator and denominator

In this, an intuitive understanding (not a proof) is given.

to understand

Consider the unit circle with angle x radians.

Length of line segment qp=sinx

length of arc rp = x

The ratio sinxx=length(qp)length(rp)

As x→0, the figure is zoomed in to the part qp and rp.

As x is getting closer to 0, the length of arc rp equals the length of line qp.

For very small values of x

sinx=x

**Limit of sin(x)/x: **

limx→0sinxx=1

problem and solutions

There are multiple proofs for limx→01-cosxx=0.

• Substitute series expansion

cosx=1-x22!+x44!+⋯

• Use the equality

1-cosx=2sin2(x2)

and use the previous result for sinxx

• Use the L'hospital's rule to differentiate numerator and denominator

In this, an intuitive understanding (not proof) is given.

to understand

Consider the unit circle with angle x radians. The length of line segment qr = 1-cosx

length of arc rp = x

The ratio 1-cosxx=length(qr)length(rp)

As x→0, the figure is zoomed in to the part qr and rp.

As x is getting closer to 0, the length of qr becomes 0 faster than the length of arc rp

For very small values of x, x is far greater than 1-cosx.

x>(1-cosx)≅0

**Limit of (1-cos(x))/x: **

limx→01-cosxx=0

example

What is limx→0tanxx?

The answer is '1'

limx→0tanxx

=limx→0sinxx×1cosx

=1×1

some results

**Limit of Logarithmic Functions: **

limx→0ln(1+x)x=1

limx→0ax-1x=lna

summary

**Limit of Exponential Functions: **

limx→∞(1+ax)x=ea

limx→0ex-1x=1

limx→∞ax=(∞ifa>1),and(0ifa<1)

limx→0(1+x)n-1x=n

limx→0sin-1xx=1

Outline

The outline of material to learn "limits (calculus)" is as follows.

Note : * click here for detailed outline of Limits(Calculus).*

→ __Indeterminate and Undefined__

→ __Indeterminate value in Functions__

→ __Expected Value__

→ __Continuity__

→ __Definition by Limits__

→ __Geometrical Explanation for Limits__

→ __Limit with Numerator and Denominator__

→ __Limits of Ratios - Examples__

→ __L'hospital Rule__

→ __Examining a function__

→ __Algebra of Limits__

→ __Limit of a Polynomial__

→ __Limit of Ratio of Zeros__

→ __Limit of ratio of infinities__

→ __limit of Binomial__

→ __Limit of Non-algebraic Functions__