Linear approximation and L'Hospital's rule

Learning objectives

Explain local linearity, that a differentiable function looks like its tangent line when viewed closely enough, and write the linearization L(x) = f(a) + f′(a)(x − a) at a point

Use the tangent line at a nearby point to approximate a function value, including values of a function known only through its derivative and one initial value

Decide whether a tangent-line approximation is an overestimate or an underestimate from the concavity of the graph near the point, and justify the decision

Verify that a limit has the indeterminate form 0/0 or ∞/∞ before applying L'Hospital's rule, and state this verification explicitly in a solution

Apply L'Hospital's rule, repeatedly if necessary, to evaluate limits of indeterminate forms, including limits that involve values read from tables or graphs of f, g, f′ and g′