Evaluate the Limit

Problem

(lim_x→0)(tan(7*x)/sin(5*x))

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which yields tan(0)/sin(0)=0/0

2. Apply the fundamental trigonometric limit (lim_θ→0)(sin(θ)/θ)=1 and its consequence (lim_θ→0)(tan(θ)/θ)=1

3. Rewrite the expression by multiplying and dividing by the appropriate factors of x to utilize these limits.

tan(7*x)/sin(5*x)=tan(7*x)/(7*x)⋅(5*x)/sin(5*x)⋅(7*x)/(5*x)

4. Simplify the expression by canceling the x terms in the final fraction.

tan(7*x)/sin(5*x)=tan(7*x)/(7*x)⋅1/sin(5*x)/(5*x)⋅7/5

5. Evaluate the limit as x→0 by applying the limit laws to each factor.

(lim_x→0)(tan(7*x)/(7*x))=1

(lim_x→0)(sin(5*x)/(5*x))=1

6. Calculate the final result by multiplying the individual limits.

1⋅1/1⋅7/5=7/5

Final Answer

(lim_x→0)(tan(7*x)/sin(5*x))=7/5

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