What is the limit as t approaches 0 of 5/t-5/(e^t-1) ?

The limit as t approaches 0 of 5/t-5/(e^t-1) is 5/2

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limit as t approaches 0 of 5/t-5/(e^t-1)

t \to 0 lim (\frac{5}{t} - \frac{5}{e ^{t} - 1})

\frac{5}{2}

Decimal Notation

2.5

Solution steps

t \to 0 lim (\frac{5}{t} - \frac{5}{e ^{t} - 1})

Simplify

\frac{5}{t} - \frac{5}{e ^{t} - 1} : \frac{5 ( e ^{t} - 1 ) - 5 t}{t ( e ^{t} - 1 )}

= t \to 0 lim (\frac{5 ( e ^{t} - 1 ) - 5 t}{t ( e ^{t} - 1 )})

Apply L'Hopital's Rule

: t \to 0 lim (\frac{5 e ^{t} - 5}{e ^{t} - 1 + e ^{t} t})

= t \to 0 lim (\frac{5 e ^{t} - 5}{e ^{t} - 1 + e ^{t} t})

Apply L'Hopital's Rule

: t \to 0 lim (\frac{5 e ^{t}}{e ^{t} t + 2 e ^{t}})

= t \to 0 lim (\frac{5 e ^{t}}{e ^{t} t + 2 e ^{t}})

Plug in the value

t = 0

= \frac{5 e ^{0}}{e ^{0} \cdot 0 + 2 e ^{0}}

Simplify

\frac{5 e ^{0}}{e ^{0} \cdot 0 + 2 e ^{0}} : \frac{5}{2}

= \frac{5}{2}

y = x^{2} - 1, x = - 2, x = 1

\frac{d}{d x} (3 (4 - 9 x)^{4})

\int \frac{7 e ^{7 x}}{1 + e ^{7 x}} d x

x \to 0 lim (\frac{tan ( 12 x )}{sin ( 4 x )})

What is the limit as t approaches 0 of 5/t-5/(e^t-1) ?
The limit as t approaches 0 of 5/t-5/(e^t-1) is 5/2