# Properties of Limits

### Notation of Limit

The limit of a function is designated by $$f\left( x \right) \to L$$ as $$x \to a$$ or using the limit notation: $$\lim\limits_{x \to a} f\left( x \right) = L.$$

Below we assume that the limits of functions $$\lim\limits_{x \to a} f\left( x \right),$$ $$\lim\limits_{x \to a} g\left( x \right),$$ $$\lim\limits_{x \to a} {f_1}\left( x \right),$$ $$\ldots,$$ $$\lim\limits_{x \to a} {f_n}\left( x \right)$$ exist.

### Sum Rule

This rule states that the limit of the sum of two functions is equal to the sum of their limits:

${\lim\limits_{x \to a} \left[ {f\left( x \right) + g\left( x \right)} \right] }={ \lim\limits_{x \to a} f\left( x \right) + \lim\limits_{x \to a} g\left( x \right).}$

### Extended Sum Rule

${\lim\limits_{x \to a} \left[ {{f_1}\left( x \right) + \ldots + {f_n}\left( x \right)} \right] } = {\lim\limits_{x \to a} {f_1}\left( x \right) + \ldots + \lim\limits_{x \to a} {f_n}\left( x \right).}$

### Constant Function Rule

The limit of a constant function is the constant:

$\lim\limits_{x \to a} C = C.$

### Constant Multiple Rule

The limit of a constant times a function is equal to the product of the constant and the limit of the function:

${\lim\limits_{x \to a} kf\left( x \right) }={ k\lim\limits_{x \to a} f\left( x \right).}$

### Product Rule

This rule says that the limit of the product of two functions is the product of their limits (if they exist):

${\lim\limits_{x \to a} \left[ {f\left( x \right)g\left( x \right)} \right] }={ \lim\limits_{x \to a} f\left( x \right) \cdot \lim\limits_{x \to a} g\left( x \right).}$

### Extended Product Rule

${\lim\limits_{x \to a} \left[ {{f_1}\left( x \right){f_2}\left( x \right) \cdots {f_n}\left( x \right)} \right] } = {\lim\limits_{x \to a} {f_1}\left( x \right) \cdot \lim\limits_{x \to a} {f_2}\left( x \right) \cdots}\kern0pt{ \lim\limits_{x \to a} {f_n}\left( x \right).}$

### Quotient Rule

The limit of quotient of two functions is the quotient of their limits, provided that the limit in the denominator function is not zero:

${\lim\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}} }={ \frac{{\lim\limits_{x \to a} f\left( x \right)}}{{\lim\limits_{x \to a} g\left( x \right)}},\;\;\;}\kern-0.3pt {\text{if}\;\;\lim\limits_{x \to a} g\left( x \right) \ne 0.}$

### Power Rule

${\lim\limits_{x \to a} {\left[ {f\left( x \right)} \right]^p} }={ {\left[ {\lim\limits_{x \to a} f\left( x \right)} \right]^p},}$

where the power $$p$$ can be any real number. In particular,

$\lim\limits_{x \to a} \sqrt[\large p\normalsize]{{f\left( x \right)}} = \sqrt[\large p\normalsize]{{\lim\limits_{x \to a} f\left( x \right)}}.$

If $$f\left( x \right) = x^n,$$ then

${\lim\limits_{x \to a} {x^n} = {a^n},\;n = 0, \pm 1, \pm 2, \ldots \;\;\;}\kern-0.3pt {\text{and}\;\;a \ne 0,\;\;\text{if}\;\;n \le 0.}$

This is a special case of the previous property.

### Limit of an Exponential Function

$\lim\limits_{x \to a} {b^{f\left( x \right)}} = {b^{\lim\limits_{x \to a} f\left( x \right)}},$

where the base $$b \gt 0.$$

### Limit of a Logarithm of a Function

${\lim\limits_{x \to a} \left[ {\log _b f\left( x \right)} \right] }={ \log_b \left[ {\lim\limits_{x \to a} f\left( x \right)} \right],}$

where the base $$b \gt 0.$$

### The Squeeze Theorem

Suppose that $$g\left( x \right) \le f\left( x \right) \le h\left( x \right)$$ for all $$x$$ close to $$a,$$ except perhaps for $$x = a.$$ If

${\lim\limits_{x \to a} g\left( x \right) = \lim\limits_{x \to a} h\left( x \right)} = {L,}$

then

$\lim\limits_{x \to a} f\left( x \right) = L.$

The idea here is that the function $$f\left( x \right)$$ is squeezed between two other functions having the same limit $$L.$$

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Find the limit $$\lim\limits_{x \to 10} \left( {2x\lg {x^3}} \right)$$.

### Example 2

Find the limit $$\lim\limits_{x \to 9} {\large\frac{{4{x^2}}}{{1 + \sqrt x }}\normalsize}$$.

### Example 3

Suppose that $$\lim\limits_{x \to 1} f\left( x \right) = 2$$ and $$\lim\limits_{x \to 1} g\left( x \right) = 3.$$ Calculate the limit $$\lim\limits_{x \to 1} {\large\frac{{g\left( x \right) – 3f\left( x \right)}}{{{f^2}\left( x \right) + g\left( x \right)}}\normalsize}.$$

### Example 4

Calculate the limit $$\lim\limits_{x \to \infty } {\large\frac{{3x + \cos x}}{{2x – 7}}\normalsize}.$$

### Example 5

Find the limit $$\lim\limits_{x \to \infty } {\large\frac{{2\sin x – 5x}}{{3x + 1}}\normalsize}.$$

### Example 1.

Find the limit $$\lim\limits_{x \to 10} \left( {2x\lg {x^3}} \right)$$.

Solution.

${\lim\limits_{x \to 10} \left( {2x\lg {x^3}} \right) } = {\lim\limits_{x \to 10} 2x \cdot \lim\limits_{x \to 10} \lg {x^3} } = {2\lim\limits_{x \to 10} x \cdot \lg \left( {\lim\limits_{x \to 10} {x^3}} \right) } = {2 \cdot 10 \cdot \lg 1000 = 20 \cdot 3 = 60. }$

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Problem 1
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Problems 2-5