Use I'Hopital's Rule more than once to rewrite the limit in its final form as lim x --> 0 lim x --> 0 e^x -sinx - 1/ x^4 + 7x^3 + 8x^2

To compute the limit

notice that when x = 0, both the numerator and denominator converge to 0. Apply l'Hopital's rule once gives

but again, this returns the indeterminate form 0/0. Applying again gives

so that the numerator converges to e⁰ + sin(0) = 1, and the denominator converges to 12•0² + 42•0 + 16 = 16.

So the limit is 1/16.

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step-by-step explanation: