helo? find limits

Question

Answer 1

(1)

The limit  is .

The above expression is in form as tends to zero.

Thus, find the limit by L-Hospital rule.

L-Hospital rule:

Suppose or , then

Apply L-Hospital rule.

Here and .

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Differentiate on each side.

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Differentiate on each side.

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Solution:

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Answer 2

(2)

The limit  is .

Consider .

To estimate the value of limit, construct a table with larger values of  by increasing rapidly.

Observe the table results,

As the value of increases, value is close to zero.

Thus, .

Solution:

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Answer 3

(3)

The limit  is .

Apply squeeze theorem:

Suppose that , where and , then  .

The range of is .

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Divide on each side by .

Apply infinite limits on each side.

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Therefore, by squeeze theorem .

Solution:

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