Question

Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow \ln 2} \frac{e^{3 x}-8}{e^{2 x}-4} $$

Short answer

The limit as \( x \) approaches \( \ln 2 \) of \( \frac{e^{3x}-8}{e^{2x}-4} \) is 3.

Step-by-step solution

Graph the Function

Using graphing utility like a graphing calculator or software such as Desmos, graph the function \( f(x) = \frac{e^{3x}-8}{e^{2x}-4} \) over a reasonable domain that certainly includes \( x = \ln 2 \). Observe that as \( x \) approaches \( \ln 2 \), \( f(x) \) heads towards a certain value. That's your visual estimate of the limit.

Use a Table to Reinforce Conclusion

Create a table with inputs approaching \( \ln 2 \) from both the left and right sides. If the table's outputs are approaching a consistent value, this further supports the estimated limit from the graph. The inputs should be values like \( \ln(2) - 0.1 , \ln(2), \ln(2) + 0.1 \). Make sure these outputs are closing in on the same value.

Find the Limit Analytically

The function can be evaluated using L'Hopital's Rule, which is valid here since the function in this form results in the indeterminate form 0/0 when \( x = \ln 2 \) is substituted into the equation. By taking the derivative of the numerator and denominator separately, the transformed function will give the exact limit: \( \lim_{x \rightarrow \ln 2} \frac{3e^{3x}}{2e^{2x}} \). From here, substitute \( x = \ln 2 \) in to find the exact value of the limit.