L'Hospital's Rule (L'Hôpital's Rule)

Mathematics · FE Reference Handbook section

Mathematics

4 formulas

10 exam-style examples

~53 min

All Mathematics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

is equal to the first of the expressions
which is not indeterminate, provided such first indicated limit exists.

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule)

Evaluate the limit L = lim(x→0) (e^{3x} − 1)/sin(5x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{3x} − 1
Denominator: sin(5x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 3e^{3x}, g'(x) = 5cos(5x)$
Substituting x = 0
$L = 3(1)/[5(1)] = 0.6000$
Numerical check at x = 0.001
$(e^{0.00300} - 1)/\sin (0.00500) = 0.6009$

Answer:

L = 3/5 = 0.6000

Why the other options are there

1.6667 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 2

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (2)

Evaluate the limit L = lim(x→0) (e^{4x} − 1)/sin(4x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{4x} − 1
Denominator: sin(4x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 4e^{4x}, g'(x) = 4cos(4x)$
Substituting x = 0
$L = 4(1)/[4(1)] = 1.0000$
Numerical check at x = 0.001
$(e^{0.00400} - 1)/\sin (0.00400) = 1.0020$

Answer:

L = 4/4 = 1.0000

Why the other options are there

1.0000 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 3

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (3)

Evaluate the limit L = lim(x→0) (e^{7x} − 1)/sin(3x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{7x} − 1
Denominator: sin(3x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 7e^{7x}, g'(x) = 3cos(3x)$
Substituting x = 0
$L = 7(1)/[3(1)] = 2.3333$
Numerical check at x = 0.001
$(e^{0.00700} - 1)/\sin (0.00300) = 2.3415$

Answer:

L = 7/3 = 2.3333

Why the other options are there

0.4286 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 4

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (4)

Evaluate the limit L = lim(x→0) (e^{4x} − 1)/sin(2x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{4x} − 1
Denominator: sin(2x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 4e^{4x}, g'(x) = 2cos(2x)$
Substituting x = 0
$L = 4(1)/[2(1)] = 2.0000$
Numerical check at x = 0.001
$(e^{0.00400} - 1)/\sin (0.00200) = 2.0040$

Answer:

L = 4/2 = 2.0000

Why the other options are there

0.5000 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 5

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (5)

Evaluate the limit L = lim(x→0) (e^{6x} − 1)/sin(6x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{6x} − 1
Denominator: sin(6x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 6e^{6x}, g'(x) = 6cos(6x)$
Substituting x = 0
$L = 6(1)/[6(1)] = 1.0000$
Numerical check at x = 0.001
$(e^{0.00600} - 1)/\sin (0.00600) = 1.0030$

Answer:

L = 6/6 = 1.0000

Why the other options are there

1.0000 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 6

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (6)

Evaluate the limit L = lim(x→0) (e^{2x} − 1)/sin(2x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{2x} − 1
Denominator: sin(2x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 2e^{2x}, g'(x) = 2cos(2x)$
Substituting x = 0
$L = 2(1)/[2(1)] = 1.0000$
Numerical check at x = 0.001
$(e^{0.00200} - 1)/\sin (0.00200) = 1.0010$

Answer:

L = 2/2 = 1.0000

Why the other options are there

1.0000 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 7

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (7)

Evaluate the limit L = lim(x→0) (e^{2x} − 1)/sin(6x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{2x} − 1
Denominator: sin(6x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 2e^{2x}, g'(x) = 6cos(6x)$
Substituting x = 0
$L = 2(1)/[6(1)] = 0.3333$
Numerical check at x = 0.001
$(e^{0.00200} - 1)/\sin (0.00600) = 0.3337$

Answer:

L = 2/6 = 0.3333

Why the other options are there

3.0000 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 8

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (8)

Evaluate the limit L = lim(x→0) (e^{3x} − 1)/sin(6x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{3x} − 1
Denominator: sin(6x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 3e^{3x}, g'(x) = 6cos(6x)$
Substituting x = 0
$L = 3(1)/[6(1)] = 0.5000$
Numerical check at x = 0.001
$(e^{0.00300} - 1)/\sin (0.00600) = 0.5008$

Answer:

L = 3/6 = 0.5000

Why the other options are there

2.0000 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 9

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (9)

Evaluate the limit L = lim(x→0) (e^{4x} − 1)/sin(6x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{4x} − 1
Denominator: sin(6x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 4e^{4x}, g'(x) = 6cos(6x)$
Substituting x = 0
$L = 4(1)/[6(1)] = 0.6667$
Numerical check at x = 0.001
$(e^{0.00400} - 1)/\sin (0.00600) = 0.6680$

Answer:

L = 4/6 = 0.6667

Why the other options are there

1.5000 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)

Example 10

Indeterminate limit by L'Hôpital's rule — L'Hospital's Rule (L'Hôpital's Rule) (10)

Evaluate the limit L = lim(x→0) (e^{4x} − 1)/sin(3x). Confirm the indeterminate form first, then apply L'Hôpital's rule and check numerically at x = 0.001.

Given

Numerator: e^{4x} − 1
Denominator: sin(3x)

Find

The limit L

Start with the thinking

At x = 0 both numerator and denominator vanish, giving the 0/0 form L'Hôpital's rule requires.
Differentiate numerator and denominator separately — never as a quotient.

Step-by-step solution

Check the form
$at x = 0: (e^{0} - 1)/\sin 0 = 0/0, so L'Hôpital's rule applies$
Formula
$L = lim f'(x)/g'(x)$
Derivatives
$f'(x) = 4e^{4x}, g'(x) = 3cos(3x)$
Substituting x = 0
$L = 4(1)/[3(1)] = 1.3333$
Numerical check at x = 0.001
$(e^{0.00400} - 1)/\sin (0.00300) = 1.3360$

Answer:

L = 4/3 = 1.3333

Why the other options are there

0.7500 (ratio inverted)
1 (assumed all 0/0 limits equal one)

Reference: FE Reference Handbook — Mathematics → L'Hospital's Rule (L'Hôpital's Rule)