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Question
`lim_(x -> 0) (15^x - 3^x - 5^x + 1)/(xtanx)` is equal to ______.
Options
`log sqrt(3) * log 5`
`log 5 * log 3`
`log sqrt(5) * log 3`
`log sqrt(1) * log 2`
MCQ
Fill in the Blanks
Solution
`lim_(x -> 0) (15^x - 3^x - 5^x + 1)/(xtanx)` is equal to `log 5 * log 3`.
Explanation:
`lim_(x -> 0) (15^x - 3^x - 5^x + 1)/(xtanx)`
= `lim_(x -> 0) ((5 xx 3)^x - 3^x - 5^x + 1)/(xtanx)`
= `lim_(x -> 0) (5^x (3^x - 1) - 1(3^x - 1))/(xtanx)`
= `lim_(x -> 0) ((5^x - 1)(3^x - 1))/(xtanx)`
= `lim_(x -> 0) (((5^x - 1)/x)*((3^x - 1)/x))/(sinx/x)`
= `log 5 * log 3`
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Limits of Exponential and Logarithmic Functions
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