Linear approximation and L'Hospital's rule

Learning objectives

Write the linearization L(x) = f(a) + f′(a)(x − a) at a point and use it to approximate nearby values of the function

Decide whether a tangent line approximation overestimates or underestimates the true value, using the concavity of the graph near the point of tangency, and justify the answer

Verify in writing that a limit has the form 0/0 or ∞/∞ before applying L'Hospital's rule, and state that the rule does not apply to any other form without rearrangement

Apply L'Hospital's rule, repeating it while the quotient remains indeterminate, and evaluate limits given only through tables or graphs of f, g, f′ and g′

Rearrange indeterminate products, differences or powers into a quotient so that L'Hospital's rule becomes applicable