Limits of Functions. Continuity

Definition

So far we have defined the concept of limit for sequences. Our next task is to extend this notion to the case of functions of real variables.

Definition
 
Assume that \( \alpha,\beta\in\mathbb R \) are real numbers and that \( f:\mathbb [\alpha, \beta]\to\mathbb R \) is a function. Let \( a\in[\alpha,\beta] \). We say that the limit \( \displaystyle\lim_{x\to a}f(x) \) exists and is equal to \( L \) if:

For every \( \varepsilon> 0 \) there exists \( \delta> 0 \) such that whenever \( x\in[\alpha,\beta] \) and \( 0< |x-a|< \delta \) the following inequality holds: \[ \left|f(x)-L\right|< \varepsilon.\]

Main properties of limits

We will skip the proofs of the following theorems. The theorems are very similar to the ones we had for sequences, and the proofs follow the same ideas.

Theorem 1 (Squeeze Theorem)
 
Assume that \( f, g,h:[\alpha,\beta]\to\mathbb R \) and that \( a\in[\alpha,\beta] \). If \( g(x)\leq f(x)\leq h(x) \) for all \( x\in[\alpha,\beta] \) and if \( \displaystyle\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L \) then \( \displaystyle \lim_{x\to a}f(x)=L \).

Theorem 2 (Limit of the sum)
 
Assume that \( a\in[\alpha,\beta] \) and that \( f,g:[\alpha,\beta]\to\mathbb R \) are two functions such that \( \displaystyle \lim_{x\to a}f(x)= F \) and \( \displaystyle \lim_{x\to a} g(x)=G \), then \( \displaystyle\lim_{x\to a}\left(f(x)+g(x)\right)=F+G \).

Theorem 3 (Limit of the product)
 
Assume that \( a\in[\alpha,\beta] \) and that \( f,g:[\alpha,\beta]\to\mathbb R \) are two functions such that \( \displaystyle \lim_{x\to a}f(x)= F \) and \( \displaystyle \lim_{x\to a} g(x)=G \), then \( \displaystyle\lim_{x\to a}\left(f(x)\cdot g(x)\right)=F\cdot G \).

Theorem 4 (Limit of the quotient)
 
Assume that \( a\in[\alpha,\beta] \) and that \( f,g:[\alpha,\beta]\to\mathbb R \) are two functions such that \( \displaystyle \lim_{x\to a}f(x)= F \), \( \displaystyle \lim_{x\to a} g(x)=G \), \( g(x)\neq 0 \) and \( G\neq 0 \), then \( \displaystyle\lim_{x\to a}\frac{f(x)}{g(x)}=\frac FG \).

Examples

Example 1
 
Prove that \( \displaystyle \lim_{x\to 0}\frac{\sin x}x=1 \).

Example 2
 
Prove that \( \displaystyle \lim_{x\to 0}\frac{1-\cos x}{x^2}=\frac12 \).

Continuity

Definition
 
A function \( f:[\alpha,\beta]\to\mathbb R \) is said to be continuous at \( a\in[\alpha,\beta] \) if \[ \lim_{x\to a}f(x)=f(a).\]

Theorem 5
 
Assume that \( f:[a,b]\to\mathbb R \) is a function that is continuous at every point of the interval \( [a,b] \). Then for every convergent sequence \( (a_n)_{n=1}^{\infty} \) whose all elements belong to \( [a,b] \) we have: \[ \lim_{n\to \infty} f(a_n)=f\left(\lim_{n\to\infty} a_n\right).\]

Theorem 6
 

The following statements hold:

  • (a) If \( f:[a,b]\to\mathbb R \) and \( g:[a,b]\to\mathbb R \) are continuous functions then so are \( f+g \), \( f\cdot g \), and \( \frac{f}{g} \) (provided that \( g\neq 0 \) in the last case).
  • (b) If \( f:[a,b]\to [c,d] \) and \( g:[c,d]\to\mathbb R \) are continuous functions then \( g\circ f:[a,b]\to\mathbb R \) is continuous.
  • (c) The following functions are continuous: \( f_m(x)=x^m \) (for \( m\in\mathbb R \) and \( x> 0 \)), \( \sin :\mathbb R\to\mathbb R \), \( \cos :\mathbb R\to\mathbb R \), \( \tan :\left(-\frac{\pi}2,\frac{\pi}2\right)\to\mathbb R \), \( \cot :\left(0,\pi\right)\to\mathbb R \), \( \exp \), \( \log: (0,+\infty)\to\mathbb R \), and their compositions and inverses.

Example 3
 
Evaluate the limit \( \displaystyle \lim_{x\to 0}\frac{\sin\left(x^2+x^3\right)}{x^2} \).


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