**Theorem. (L’Hospital’s Rule, 0/0 form).** Suppose f and g are differentiable and g’(x) ≠ 0 on an open interval (a, b) containing c (except possibly at c). Suppose that

where L is a a real number, ∞, or -∞. Then

** Remark**. This theorem is valid for one-sided limits as well as the two sided limit.

This theorem is also true if c = ∞, or c = -∞.

**Theorem. (L’Hospital’s Rule, ∞/∞form)**. The previous theorem is also valid for the case when

**Exercise 1**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form 0/0. We compute this limit applying L’Hospital’s Rule as follows:

**Exercise 2**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form 0/0. We compute this limit applying L’Hospital’s Rule as follows:

**Exercise 3**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form 0/0. We compute this limit applying L’Hospital’s Rule as follows:

**Exercise 4**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form 0/0. We compute this limit applying L’Hospital’s Rule as follows:

**Exercise 5.** Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form 0/0. We compute this limit applying L’Hospital’s Rule as follows:

Applying again L’Hospital’s Rule we have:

**Exercise 6**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form ∞/∞. We compute this limit applying L’Hospital’s Rule as follows:

Applying again L’Hospital’s Rule we have:

**Exercise 7**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form 0·∞ when x tends to 0. Then, before applying L’Hospital’s Rule, we have to rig the assigned function to make an indeterminate quotient as follows:

and the limit becomes:

Now this limit yields the indeterminate form ∞ /∞ when x tends to 0. Then, we compute this limit applying L’Hospital’s Rule as follows:

**Exercise 8**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form ∞ – ∞ when x tends to 0. Then, before applying L’Hospital’s Rule, we have to rig the assigned function to make an indeterminate quotient as follows:

and the limit becomes:

Now this limit yields the indeterminate form 0/0 when x tends to 0. Then, we compute this limit applying L’Hospital’s Rule as follows:

Applying again L’Hospital’s Rule we have:

**Exercise 9**. Find the following limit using L’Hospital’s Rule

*Solution*. The assigned function yields the indeterminate form 1^{∞} when x tends to 0^{+}. Then, before applying L’Hospital’s Rule, we have to rig the assigned function to make an indeterminate quotient as follows:

and the limit becomes:

where

Now this limit yields the indeterminate form 0/0 when x tends to 0. So, we can compute this limit applying L’Hospital’s Rule as follows:

where

*1*. Functions and their graph. One–one and onto functions. Composite and inverse functions.

*2*. Linear inequalities. Systems of linear inequalities

*3*. Polynomial inequalities. Rational inequalities. Absolute-value inequalities

*6*. Differentiation. Derivative rules, the chain rule.

*7*. Application of differentiation: L'Hospital's Rule

*8*. Vertical, Horizontal and Slant asymptotes

*9*. Higher Order Derivatives. Applications of differentiation: local and absolute extremes of a function

*10*. Curve Sketching

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