Theorems of Limits of functions

Theorems of Limits of functions

By SLSI Calculus

Theorem 1

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The limit of a constant as x approaches to a is the constant itself.

1-2-3​

Theorem 1

Multiple Choice

$$\lim_{x\rightarrow3}\ -4\ =\$$  

• 3

• -4

• 4

• -3

Multiple Choice

$$\lim_{y\rightarrow5}6\ =\$$  

• 5

• -5

• 6

• -6

Multiple Choice

$$\lim_{x\rightarrow10}1\ =\$$  

• 1

• -1

• 10

• -10

The limit of the identity function as x approaches a is a. This may be thought of as the substitution law, because x is simply substituted by a.

1-2-3​

Theorem 2

Multiple Choice

$$\lim_{x\rightarrow-3}x$$  

• -3

• 3

• x

• 0

Multiple Choice

$$\lim_{x\rightarrow5}\left(-x\right)\ =\$$  

• 5

• -5

• x

• 0

Multiple Choice

$$\lim_{x\rightarrow\left(-2\right)}\left(-x\right)\ =\$$  

• -2

• 2

• x

• -x

The limit of a constant multiple c times a function as x approaches a is equal to the constant multiple c times the limit of the function provided that limit of functions exists.

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Theorem 3

Multiple Choice

$$Detre\min e\ if\ the\ solution\ is\ true\ or\ false.\ \lim_{x\rightarrow5}2x\ =2\lim_{x\rightarrow5}x=2\left(5\right)\ =\ 10\$$  

• TRUE

• FALSE

Multiple Choice

$$\lim_{x\rightarrow-1}6x\ =-1\lim_{x\rightarrow-1}6=-1\left(6\right)\ =\ -6\$$  

• TRUE

• FALSE

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The limit of the sum of the two functions is equal to the sum of the individual limit provided that the limit of each function as x approaches a exists.

Theorem 4

The limit of the difference of two given functions is equal to the difference of the individual limit provided that the limit of each function as x approaches a exists.

Theorem 5

The limit of the difference of two given functions is equal to the difference of the individual limit provided that the limit of each function as x approaches a exists.

Theorem 6

The limit of the quotient of the two functions is equal to the quotient of the individual limit provided that the limit of the divisor is not equal to 0 and the limit of each function exists.

Theorem 7

Multiple Choice

$$\lim_{x\rightarrow2}\left(x+5\right)\ =\ \lim_{x\rightarrow2}x\ +\ \lim_{x\rightarrow2}5\ =\left(2\right)\left(5\right)\ =\ 7$$  

• TRUE

• FALSE

Multiple Choice

$$\lim_{x\rightarrow2}\left(x-5\right)\ =\ \lim_{x\rightarrow2}x\ -\ \lim_{x\rightarrow2}5\ =2-5\ =\ -3$$  

• TRUE

• FALSE

Multiple Choice

$$\lim_{x\rightarrow2}\left[4\left(x-5\right)\right]\ =\ \left[\lim_{x\rightarrow2}4\left(\lim_{x\rightarrow2}x\ -\ \lim_{x\rightarrow2}5\ \right)\right]=4\left(2-5\right)\ =\ -12$$  

• TRUE

• FALSE

The limit of the nth power of a function is equal to the nth power of the limit of that function provided that n is positive integer and the limit of f(x) as x approaches a exists.

Theorem 8

The limit of the nth power of a function is equal to the nth power of the limit of that function provided that n is positive integer and the limit of f(x) as x approaches a exists.

Theorem 8

Theorem 9

The limit of the nth root of a function is equal to the principal nth root of the limit of that function provided that n is a positive integer and the limit of the function is positive if n is even.

Multiple Choice

$$\lim_{x\rightarrow20}\sqrt[]{x+5}\ =\$$  

• 5

• 25

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