Limit Problems

Solutions

1Using the definition of a limit, prove that:

2Using the graph of the function f(x), determine the following limits.

Limits

3Using the definition of a limit, prove that:

Limit Problem    has limit 1 as x tiende 0

Calculate the Following Limits:

4Calculate the Limit

5Calculate the Limit

6Calculate the Limit

7Calculate the Limit


1

Using the definition of a limit, prove that:

Limit Operations

Limit Operations

Limit Operations

Limit Operations

To check this, take a ε = 0.01.

Limit Solutions

Limit Operations

For x = 0.995    f(x) = (0.995 + 3)/2 = 1.9975.

For x = 1.015    f(x) = (1.015 + 3)/2 = 2.0075.


2

Using the graph of the function f(x), determine the following limits.

Limits

Calculate the Limit

Calculate the Limit

Calculate the Limit

Calculate the Limit

Calculate the Limit


3

Using the definition of a limit, prove that:

Calculate the Limit    has limit −1 as x tiende 0

Limit Operations

Left side limit.

Limit Operations

Limit Operations

Limit Operations

Limit Operations

Right side limit

Limit Operations

Limit Operations

Limit Operations

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Limit Solution

4

Calculate the limit:

Calculate the Limit

Limit Operations

Limit Operations

Calculate the side limits to determine the sign of ∞.

Limit Operations

Limit Operations

No limit.


5

Calculate the limit:

Calculate the Limit

Limit Operations

Limit Solution


6

Calculate the limit:

Calculate the Limit

Limit Operations

Limit Operations

Limit Operations

Limit Solution


7

Calculate the limit:

Calculate the Limit

Limit Operations

Limit Operations

Limit Solution