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Exercise 1

Using the definition of a limit, prove that:

$\text{[math]}$

Exercise 2

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

Exercise 3

Using the definition of a limit, prove that:

$\text{[math]}$ has a limit −1 as x 0

Calculate the Following Limits:

Exercise 4

$\text{[math]}$

Exercise 5

$\text{[math]}$

Exercise 6

$\text{[math]}$

Exercise 7

$\text{[math]}$

Solution of exercise 1

Using the definition of a limit, prove that:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

If $\text{[math]}$

$\text{[math]}$

To check this, take a $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$

For $\text{[math]}$ .

Solution of exercise 2

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

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Solution of exercise 3

Using the definition of a limit, prove that:

$\text{[math]}$ has a limit −1 as x 0

$\text{[math]}$

Left side limit.

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Right side limit

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Solution of exercise 4

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

Calculate the side limits to determine the sign of $\text{[math]}$ .

$\text{[math]}$ $\text{[math]}$

No limit.

Solution of exercise 5

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

Solution of exercise 6

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$

Solution of exercise 7

Calculate the limit:

$\text{[math]}$

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Summarise with AI:

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Limit Problems

Exercise 1

Exercise 2

Exercise 3

Calculate the Following Limits:

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

Solution of exercise 7

Theory

Calculating Limits

Continuity

Continuous Function

Continuity on a Closed Interval

Discontinuity

Essential Discontinuity

Intermediate Value Theorem

Jump Discontinuity

One-Sided Limit

Zero Times Infinity

Infinity Minus Infinity

Infinity over Infinity

Extreme Value Theorem

Bolzano’s Theorem

Limits at Infinity

Infinite Limit

Limit of a Logarithmic Function

Removable Discontinuity

Limit of an Exponential Function

Limit Rules

Properties of Infinity

One to the Power of Infinity

Limits and Dividing by Zero: Theory and Examples

Understanding All the Indeterminate Forms

Zero Over Zero: Indeterminate Forms and L’Hôpital’s Rule

Formulas

Continuity Formulas

Limit Formulas

Exercises

Limit Problems

Intermediate Value Theorem Problems

Continuity Worksheet

Continuity Practice Problems for A Level Maths

Limits Worksheet With Worked Solutions

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