# Calculus 1 : How to find the meaning of functions

## Example Questions

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### Example Question #1 : How To Find The Meaning Of Functions

Take the limit

Explanation:

First, multiply the numerator and denominator by and it turns into

Factor the numerator and then cancel out the 'x+2'

After taking the limit, the answer is

### Example Question #2 : How To Find The Meaning Of Functions

If this limit is true, then what is the value of 'a'?

Explanation:

Factor the numerator

Cancel the , plug in the limit and then solve for 'a'

### Example Question #3 : How To Find The Meaning Of Functions

We have a line described as . Find the minimum distance between the origin and a point on that line.

Explanation:

We have the origin  and a point  located on the line.  That point represents the minimum distance to the orgin.  Apply the distance formula to these two points,

Plug in the line equation, take the derivative, set it equal to zero, and solve for x.

Use this  value to find

So we have the point , which is closest to the origin. We can now find its distance from that origin.

### Example Question #4 : How To Find The Meaning Of Functions

We have the following,

What is c?

Explanation:

First, factor the numerator of the integrand.

Cancel out

Perform the integral and then solve for

### Example Question #5 : How To Find The Meaning Of Functions

If , find

Explanation:

Taking the derivative of an integral yields the original function, but because we have a different variable in the integration limits, the variable switches

### Example Question #6 : How To Find The Meaning Of Functions

Evaluate

Explanation:

using integration identities:

### Example Question #7 : How To Find The Meaning Of Functions

Evaluate

Explanation:

evaluate at

### Example Question #8 : How To Find The Meaning Of Functions

Evaluate

Explanation:

Intergation by substitution

new endpoints:

New Equation:

at

### Example Question #9 : How To Find The Meaning Of Functions

What is  ?

0

-1

undefined

1

0

Explanation:

The relationship between  and x is an exponential relationship;  is going to  exponentially faster than  is going to  . One way to prove this is to write  and use L'Hôpital's rule:

### Example Question #10 : How To Find The Meaning Of Functions

Where is  discontinuous? Are those discontinuities removable?

Removable discontinuities at and .

Removable discontinuity at , essential discontinuity at .

Essential discontinuities at  and .

Removable discontinuity at , essential discontinuity at

Removable discontinuities at  and

Removable discontinuity at , essential discontinuity at

Explanation:

The rational function   has a denominator with two roots,  and . These are discontinuities.

Factoring both top and bottom and canceling a term  tells us that this function is equal to

except at . This point is a removable discontinuity.  is therefore an essential discontinuity where the ratio goes to .

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