Evaluate the Limit limit as x approaches 0 of (tan(5x))/x

Problem

(lim_x→0)(tan(5*x)/x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression.

tan(5*(0))/0=0/0

2. Rewrite the expression to use the fundamental trigonometric limit (lim_θ→0)(sin(θ)/θ)=1 by expressing tan(5*x) as sin(5*x)/cos(5*x)

(lim_x→0)(sin(5*x)/(x*cos(5*x)))

3. Manipulate the fraction to create the form sin(5*x)/(5*x) by multiplying the numerator and denominator by 5

(lim_x→0)(5)⋅sin(5*x)/(5*x)⋅1/cos(5*x)

4. Apply the limit laws to evaluate each part of the product separately.

5⋅((lim_x→0)(sin(5*x)/(5*x)))⋅((lim_x→0)(1/cos(5*x)))

5. Substitute the known limits (lim_u→0)(sin(u)/u)=1 and cos(0)=1

5⋅1⋅1/1=5

Final Answer

(lim_x→0)(tan(5*x)/x)=5

Want more problems? Check here!