Limits and Continuity

The Fundamental Sine Limit

Consider the following limit:

\lim_{x\to0}\frac{\sin(x)}{x}

We can quickly see that this is a “ $0/0$ ” limit, and therefore is indeterminate. We can't factor anything or otherwise simplify the expression to get rid of the issue.

At the initial settings this is the original sine-ratio graph. Adjust the coefficient and compare angle units. The hole remains at zero.

\displaystyle y=\frac{\sin(1x)}{x},\quad x\ne0

Coefficient k1Angle unit Closeness · larger means nearer1

\displaystyle h=10^{-1}=0.1

Approach	$\displaystyle x$	$\displaystyle f(x)$
From the left	-0.1	0.99833417
From the right	0.1	0.99833417

\displaystyle \lim_{x\to0}\frac{\sin(1x)}x=1

The ratio is undefined at x = 0. A limit describes nearby values, not substitution at zero.

It is pretty clear from the graph that the ratio approaches 1. Numerical values and a graph help us see the behavior; they are not by themselves a proof.

THEOREM

The Fundamental Sine Limit

With angles measured in radians,

\lim_{x\to0}\frac{\sin(x)}{x}=\lim_{x\to0}\frac{x}{\sin(x)}=1

To see why, for $0<x<\dfrac\pi2$ , unit-circle geometry gives $\sin x<x<\tan x$ , and hence

\cos x<\frac{\sin x}{x}<1

As $x\to0^+$ , both bounds approach 1. Since $\dfrac{\sin(-x)}{-x}=\dfrac{\sin x}{x}$ , the left-hand limit is the same. Taking the reciprocal gives the second limit in the theorem.

QUESTION

In radians, what is

\displaystyle \displaystyle\lim_{x\to0}\dfrac{\sin(3x)}x

Evaluate the following limits. All angles are in radians.

(a) $\displaystyle \lim\limits_{t\to0}\dfrac{5\sin(t)}{2t}$

(b) $\displaystyle \lim\limits_{\theta\to0}\dfrac{\theta}{\sin(4\theta)}$

(d) $\displaystyle \lim\limits_{x\to0}\dfrac{\sin(3x)}{\sin(2x)}$

(e) $\displaystyle \lim\limits_{\theta\to0}\dfrac{100\theta}{\tan(3\theta)}$

(f) $\displaystyle \lim\limits_{h\to0}\dfrac{\cos(h)-1}{h}$

◀ Previous SectionEvaluating Indeterminate Limits Next Section ▶Limits at Infinity

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Limits and Continuity

The Fundamental Sine Limit

Consider the following limit:

\lim_{x\to0}\frac{\sin(x)}{x}

We can quickly see that this is a “ $0/0$ ” limit, and therefore is indeterminate. We can't factor anything or otherwise simplify the expression to get rid of the issue.

At the initial settings this is the original sine-ratio graph. Adjust the coefficient and compare angle units. The hole remains at zero.

\displaystyle y=\frac{\sin(1x)}{x},\quad x\ne0

Coefficient k1Angle unit Closeness · larger means nearer1

\displaystyle h=10^{-1}=0.1

Approach	$\displaystyle x$	$\displaystyle f(x)$
From the left	-0.1	0.99833417
From the right	0.1	0.99833417

\displaystyle \lim_{x\to0}\frac{\sin(1x)}x=1

The ratio is undefined at x = 0. A limit describes nearby values, not substitution at zero.

It is pretty clear from the graph that the ratio approaches 1. Numerical values and a graph help us see the behavior; they are not by themselves a proof.

THEOREM

The Fundamental Sine Limit

With angles measured in radians,

\lim_{x\to0}\frac{\sin(x)}{x}=\lim_{x\to0}\frac{x}{\sin(x)}=1

To see why, for $0<x<\dfrac\pi2$ , unit-circle geometry gives $\sin x<x<\tan x$ , and hence

\cos x<\frac{\sin x}{x}<1

As $x\to0^+$ , both bounds approach 1. Since $\dfrac{\sin(-x)}{-x}=\dfrac{\sin x}{x}$ , the left-hand limit is the same. Taking the reciprocal gives the second limit in the theorem.

QUESTION

In radians, what is

\displaystyle \displaystyle\lim_{x\to0}\dfrac{\sin(3x)}x

Evaluate the following limits. All angles are in radians.

(a) $\displaystyle \lim\limits_{t\to0}\dfrac{5\sin(t)}{2t}$

(b) $\displaystyle \lim\limits_{\theta\to0}\dfrac{\theta}{\sin(4\theta)}$

(d) $\displaystyle \lim\limits_{x\to0}\dfrac{\sin(3x)}{\sin(2x)}$

(e) $\displaystyle \lim\limits_{\theta\to0}\dfrac{100\theta}{\tan(3\theta)}$

(f) $\displaystyle \lim\limits_{h\to0}\dfrac{\cos(h)-1}{h}$

◀ Previous SectionEvaluating Indeterminate Limits Next Section ▶Limits at Infinity

The Fundamental Sine Limit

The Fundamental Sine Limit

Why radians matter

The Fundamental Sine Limit

KNOWLEDGE QUIZ

Use the fundamental sine limit

The Fundamental Sine Limit

The Fundamental Sine Limit

Why radians matter

The Fundamental Sine Limit

KNOWLEDGE QUIZ

Use the fundamental sine limit