Evaluate Using L'Hospital's Rule limit as x approaches 8 of (1-5/x)^x

Problem

(lim_x→∞)(1−5/x)

Solution

1. Identify the indeterminate form. As x→∞ the base (1−5/x)→1 and the exponent x→∞ resulting in the indeterminate form 1

2. Rewrite the expression using the natural logarithm and exponential function to prepare for L'Hospital's Rule. Let y=(1−5/x)x then ln(y)=x*ln(1−5/x)

3. Transform the product into a quotient to create a 0/0 form.

(lim_x→∞)(ln(y))=(lim_x→∞)(ln(1−5/x)/1/x)

4. Apply L'Hospital's Rule by differentiating the numerator and the denominator with respect to x

d(ln(1−5*x(−1)))/d(x)=1/(1−5/x)⋅5/(x2)

d(x(−1))/d(x)=−1/(x2)

5. Simplify the resulting limit expression.

(lim_x→∞)(5/(x2*(1−5/x))/(−1/(x2)))=(lim_x→∞)((−5)/(1−5/x))

6. Evaluate the limit of the logarithm.

(lim_x→∞)((−5)/(1−0))=−5

7. Exponentiate the result to find the limit of the original function. Since (lim_)(ln(y))=−5 then (lim_)(y)=e(−5)

Final Answer

(lim_x→∞)(1−5/x)=e(−5)

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