#### To determine

**(a)**

What does 1’ Hospital’s Rule say.

**Solution:** To evaluate limits involving indeterminate forms. limx→cf(x)g(x)=limx→cf'(x)g'(x).

**Given:** 1’ Hospital Rule.

#### Explanation

L’ Hospital’s rule uses derivative to help evaluate limits involving indeterminate forms.

L’s Hospital rule states that for function *f* and *g* which are differentiable on an open interval *I,* expect possibly at a point *c* contained in *I*. If:

limx→cf(x)=limx→cg(x)=0 or ±∞.

g'(x)≠0

For all x in I with x≠c and

limx→cf'(x)g'(x) Exist than

limx→cf(x)g(x)=limx→cf'(x)g'(x)

**Conclusion:** Hence, limx→cf(x)g(x)=limx→cf'(x)g'(x).

**(b)**

**To determine:** How can you use 1-Hospital’s rule if you have a product f(x)⋅g(x), where f(x)→0 and g(x)→∞ and x→a?

**Solution:** This by changing multiplication into division by the reciprocal.

**Given: **f(x)→0,g(x)→∞ and x→a.

Take an example

limx→∞x⋅sin(1x)

Put the limit get 0⋅∞ form and apply L’s Hospital rule to get:

=limx→∞sin(1x)1x=limx→∞−1x2cos(1x)−1x2

=limx→∞cos(1x)1=cos0=1

**Conclusion:** Hence, by changing multiplication into division by the reciprocal.

**(c)**

**To determine:** How can we you use 1’s Hospital rule if you have a difference f(x)−g(x), where f(x)→0 and g(x)→0 as x→a.

**Solution:** Take common in function so it will convert into simple form.

**Given:** f(x)−g(x),f(x)→0,g(x)→0 and x→a.

Let limx→0(x2−x)

=limx→0x(x−1)=limx→0x−11x

Put the value of limit so get

=limx→0x−11x=−1∞=0

**Conclusion:** Hence, in case of 0 form of function, take the common in function and then it will convert into simple form.

**(d)**

**To determine:** How can you use l’s Hospital’s rule if you have a power [f(x)]g(x), where f(x)→0 and g(x)→0 as x→a?

**Solution:** Use the natural log to undo the power so it will convert into (∞∞) form.

**Given:** [f(x)]g(x),f(x)→0,g(x)→0 and x→a.

Let

y=limx→0+(2x)xlny=ln(limx→0+(2x)x)=limx→0+ln(2x)x=limx→0+ln2x1x

lny=limx→0+22x−1x2=limx→0+(−x)=0y=1

**Conclusion:** Hence, by use the natural log to undo the power, it will convert into (∞∞) form.