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L'Hopital's Rule with exponential denominator

Evaluate the limit lim⁡x→2x2−4ex−e2.\lim_{x \to 2} \frac{x^2 - 4}{e^x - e^2}. First confirm that direct substitution produces an indeterminate form, then apply the appropriate technique to find the limit's value.

Hint

Check that plugging in x=2x=2 gives 0/00/0. Since this is an indeterminate form, differentiate the numerator and denominator separately (do not use the quotient rule), then substitute x=2x=2 into the new expression.

Step-by-step solution

Substituting x=2x=2 gives numerator 22−4=02^2-4=0 and denominator e2−e2=0e^2-e^2=0, so the limit has the indeterminate form 0/00/0. This confirms that L'Hopital's Rule applies.

Differentiate numerator and denominator separately: ddx(x2−4)=2x\frac{d}{dx}(x^2-4)=2x and ddx(ex−e2)=ex\frac{d}{dx}(e^x-e^2)=e^x. So the limit equals lim⁡x→22xex=4e2\lim_{x\to 2} \frac{2x}{e^x} = \frac{4}{e^2}.

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