Find the limit if it exists.
$\lim _{x \rightarrow 9} 6(x-8)^{\frac{1}{x-9}}$
Answer (2)
Recognize the indeterminate form and rewrite the expression using exponentials and logarithms.
Apply the substitution u = x − 9 to simplify the limit.
Use L'Hopital's rule to evaluate the limit of the exponent.
Calculate the final limit: 6 e .
Explanation
Problem Setup We are asked to find the limit of the function 6 ( x − 8 ) x − 9 1 as x approaches 9.
Indeterminate Form The limit is of the form 6 ( 9 − 8 ) 9 − 9 1 = 6 ( 1 ) 0 1 , which is an indeterminate form. To solve this, we can rewrite the expression using the exponential function and logarithm.
Rewriting the Expression We rewrite the expression as 6 e l n (( x − 8 ) x − 9 1 ) = 6 e x − 9 1 l n ( x − 8 ) . Now we need to find the limit of the exponent x − 9 l n ( x − 8 ) as x approaches 9.
Substitution Let u = x − 9 , then x = u + 9 and x − 8 = u + 1 . The limit becomes lim u → 0 u l n ( u + 1 ) .
L'Hopital's Rule We can use L'Hopital's rule to evaluate the limit: lim u → 0 u l n ( u + 1 ) = lim u → 0 1 u + 1 1 = lim u → 0 u + 1 1 = 1 .
Final Calculation Therefore, the limit of the original expression is 6 e 1 = 6 e .
Examples
In thermodynamics, when analyzing the behavior of gases under varying conditions, limits of exponential functions are crucial. For example, understanding how pressure changes with infinitesimal changes in volume involves evaluating limits similar to the one in this problem. This helps engineers design efficient engines and predict gas behavior accurately.
The limit lim x → 9 6 ( x − 8 ) x − 9 1 evaluates to 6 e . This is found by rewriting the expression using exponentials and applying L'Hopital's Rule to resolve the indeterminate form. Ultimately, the limit exists and equals 6 e .
;
