10.3 More on Limits

494 days ago by mpaul

Informal Definition of Limits

\lim_{x \rightarrow a} f(x) = L means that f(x) gets arbitrarily close to L as x gets arbitrarily close to a.

Seven Properties of Limits

Following are three variables f, g, and c:
f(x) = 2*x - 1 g(x) = x^2 - 1 c = 100 f;g;c 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}100
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}100

f and g refer to functions, and c refers to a constant.

You can change them to any other values you'd like as you work through the examples.

1.  \lim_{x \rightarrow c}(f(x) + g(x)) = \lim_{x \rightarrow c}f(x) + \lim_{x \rightarrow c}g(x)

     The limit of a sum is the sum of the limits.
f; g; f+g 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} + 2 \, x - 2
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} + 2 \, x - 2

\lim_{x \rightarrow c} f(x), \lim_{x \rightarrow c} g(x):
lim(f, x=c); lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999

\lim_{x \rightarrow c}(f + g)(x)
f+g; lim(f+g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} + 2 \, x - 2
\newcommand{\Bold}[1]{\mathbf{#1}}100
\newcommand{\Bold}[1]{\mathbf{#1}}10198
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} + 2 \, x - 2
\newcommand{\Bold}[1]{\mathbf{#1}}100
\newcommand{\Bold}[1]{\mathbf{#1}}10198

\lim_{x=c}f(x) + \lim_{x=c}g(x)
lim(f, x=c) + lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}10198
\newcommand{\Bold}[1]{\mathbf{#1}}10198

2.  \lim_{x \rightarrow c}(f(x) - g(x)) = \lim_{x \rightarrow c}f(x) - \lim_{x \rightarrow c}g(x)

     The limit of a difference is the difference of the limits.
f; g; f-g 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -x^{2} + 2 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -x^{2} + 2 \, x

\lim_{x \rightarrow c} f(x), \lim_{x \rightarrow c} g(x):
lim(f, x=c); lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999

\lim_{x \rightarrow c}(f - g)(x)
f-g; lim(f-g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -x^{2} + 2 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}-9800
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -x^{2} + 2 \, x
\newcommand{\Bold}[1]{\mathbf{#1}}-9800

\lim_{x=c}f(x) - \lim_{x=c}g(x)
lim(f, x=c) - lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}-9800
\newcommand{\Bold}[1]{\mathbf{#1}}-9800

3.  \lim_{x \rightarrow c}(f(x) \cdot g(x)) = \lim_{x \rightarrow c}f(x) \cdot \lim_{x \rightarrow c}g(x)

     The limit of a product is the product of the limits.
f; g; f*g 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ {\left(2 \, x - 1\right)} {\left(x^{2} - 1\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ {\left(2 \, x - 1\right)} {\left(x^{2} - 1\right)}

\lim_{x \rightarrow c} f(x), \lim_{x \rightarrow c} g(x):
lim(f, x=c); lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999

\lim_{x \rightarrow c}(f \cdot g)(x)
f*g; lim(f*g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ {\left(2 \, x - 1\right)} {\left(x^{2} - 1\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}1989801
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ {\left(2 \, x - 1\right)} {\left(x^{2} - 1\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}1989801

\lim_{x=c}f(x) \cdot \lim_{x=c}g(x)
lim(f, x=c) * lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}1989801
\newcommand{\Bold}[1]{\mathbf{#1}}1989801

4.  \lim_{x \rightarrow c}(k \cdot g(x)) = k \cdot \lim_{x \rightarrow c}g(x)

     The limit of a scaling is the scaling of the limit.

Here we're going to introduce another variable k to represent a scalar.
k = 4 
       

\lim_{x \rightarrow c} g(x)
g; lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}9999
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}9999

\lim_{x \rightarrow c}k \cdot g(x)
k*g; lim(k*g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 4 \, x^{2} - 4
\newcommand{\Bold}[1]{\mathbf{#1}}39996
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 4 \, x^{2} - 4
\newcommand{\Bold}[1]{\mathbf{#1}}39996

k\lim_{x \rightarrow c}g(x)
k*lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}39996
\newcommand{\Bold}[1]{\mathbf{#1}}39996

5.  \lim_{x \rightarrow c}\frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow c}f(x)}{\lim_{x \rightarrow c}g(x)}

     The limit of a ratio is the ratio of the limits.
f; g; f/g 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{2 \, x - 1}{x^{2} - 1}
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ x^{2} - 1
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{2 \, x - 1}{x^{2} - 1}

\lim_{x \rightarrow c} f(x), \lim_{x \rightarrow c} g(x):
lim(f, x=c); lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}9999

\lim_{x \rightarrow c}(\frac{f}{g})(x)
f/g; lim(f/g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{2 \, x - 1}{x^{2} - 1}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{199}{9999}
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{2 \, x - 1}{x^{2} - 1}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{199}{9999}

\frac{\lim_{x=c}f(x)}{ \lim_{x=c}g(x)}
lim(f, x=c)/lim(g, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{199}{9999}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{199}{9999}

6.  lim_{x \rightarrow c} (f(x))^n = (lim_{x \rightarrow c} f(x))^n

     The limit of a power is the power of the limit.
n = 5 
       

\lim_{x \rightarrow c} (f(x))^n
f^n; lim(f^n, x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ {\left(2 \, x - 1\right)}^{5}
\newcommand{\Bold}[1]{\mathbf{#1}}312079600999
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ {\left(2 \, x - 1\right)}^{5}
\newcommand{\Bold}[1]{\mathbf{#1}}312079600999

(\lim_{x \rightarrow c}f(x))^n
lim(f, x=c); (lim(f, x=c))^n 
        
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}312079600999
\newcommand{\Bold}[1]{\mathbf{#1}}199
\newcommand{\Bold}[1]{\mathbf{#1}}312079600999

7.  \lim_{x \rightarrow c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \rightarrow c} f(x)}

     The limit of a root is the root of the limit.

\lim_{x \rightarrow c} \sqrt[n]{f(x)}
f; n; lim(f(x)^(1/n), x=c) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}5
\newcommand{\Bold}[1]{\mathbf{#1}}199^{\frac{1}{5}}
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ 2 \, x - 1
\newcommand{\Bold}[1]{\mathbf{#1}}5
\newcommand{\Bold}[1]{\mathbf{#1}}199^{\frac{1}{5}}

\sqrt[n]{\lim_{x \rightarrow c} f(x)}
lim(f(x), x=c)^(1/n) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}199^{\frac{1}{5}}
\newcommand{\Bold}[1]{\mathbf{#1}}199^{\frac{1}{5}}

Limits at Infinity

lim_{x \rightarrow \infty} f(x) = L means that f(x) gets arbitrarily close to L as x gets arbitrarily large.

lim_{x \rightarrow -\infty} f(x) = L means that f(x) gets arbitrarily close to L as - x gets arbitrarily large.
f(x) = (x+1)^2/(x^2-3) f 
        
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{{\left(x + 1\right)}^{2}}{x^{2} - 3}
\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ \frac{{\left(x + 1\right)}^{2}}{x^{2} - 3}
plot(f, -10, 10, ymax = 5, ymin = -5) 
       
f(1000) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1002001}{999997}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1002001}{999997}
lim(f, x=oo) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}1
lim(f, x=-oo) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}1
 
       

\lim_{x \rightarrow 0} \frac{\sin{x}}{x} = 1
lim(sin(x)/x, x=0) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}1
plot(sin(x)/x, -10, 10) + point((0,1), color = 'red') 
       

This is a useful limit in the study of Calculus.

Here is an elegant proof of it:

Imagine an n-sided regular polygon inscribed in a circle of radius r.

The polygon is composed of \frac{2\pi}{x} isosceles triangles where the vertex angle = x.

The area of one of these triangles is \frac{1}{2} r^2 \sin{x}, and

the area of all of these triangles is \frac{2\pi}{x} \cdot \frac{1}{2} r^2 \sin{x}.

As the angle x decreases, the number of triangles will increase,

and the sum of their areas will get arbitrarily close to the area of the circle.

\frac{2\pi}{x} \cdot \frac{1}{2} r^2 \sin{x} simplifies quite nicely to \pi r^2 \frac{\sin x}{x}.

Therefore, \lim_{x \rightarrow 0} \pi r^2 \frac{\sin x}{x} = \pi r^2,

and \pi r^2 \cdot \lim_{x \rightarrow 0}\frac{\sin x}{x} = \pi r^2,

proving that \lim_{x \rightarrow 0} \frac{\sin x}{x} = 1.