Evaluate lim_{x->π/4} (tan x - 1)/(x - π/4).

Evaluate lim_{x->π/4} (tan x - 1)/(x - π/4).

• A. 0
• B. 1
• C. 2
• D. -2

Answer: (C)

This is the difference quotient for tan x at the point x = π/4. In other words, it’s the derivative of tan at a = π/4. Since the derivative of tan x is sec^2 x, evaluate at π/4: sec^2(π/4) = 1/cos^2(π/4) = 1/(√2/2)^2 = 2. So the limit is 2. An alternative view uses the identity tan x − tan a = sin(x − a)/(cos x cos a). Then the expression becomes sin(x − a)/[(x − a) cos x cos a], which tends to 1/(cos a cos a) = 1/cos^2 a = 2 as x → a.

This is the difference quotient for tan x at the point x = π/4. In other words, it’s the derivative of tan at a = π/4. Since the derivative of tan x is sec^2 x, evaluate at π/4: sec^2(π/4) = 1/cos^2(π/4) = 1/(√2/2)^2 = 2. So the limit is 2.

An alternative view uses the identity tan x − tan a = sin(x − a)/(cos x cos a). Then the expression becomes sin(x − a)/[(x − a) cos x cos a], which tends to 1/(cos a cos a) = 1/cos^2 a = 2 as x → a.