Calculus Examples

$lim_{t \to 0} \frac{1}{t} - \frac{1}{t^{2} + t}$

Step 1

Simplify the limit argument.

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Step 1.1

Combine terms.

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Step 1.1.1

To write $\frac{1}{t}$ as a fraction with a common denominator, multiply by $\frac{t^{2} + t}{t^{2} + t}$ .

$lim_{t \to 0} \frac{1}{t} \cdot \frac{t^{2} + t}{t^{2} + t} - \frac{1}{t^{2} + t}$

Step 1.1.2

To write $- \frac{1}{t^{2} + t}$ as a fraction with a common denominator, multiply by $\frac{t}{t}$ .

$lim_{t \to 0} \frac{1}{t} \cdot \frac{t^{2} + t}{t^{2} + t} - \frac{1}{t^{2} + t} \cdot \frac{t}{t}$

Step 1.1.3

Write each expression with a common denominator of $t (t^{2} + t)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.3.1

Multiply $\frac{1}{t}$ by $\frac{t^{2} + t}{t^{2} + t}$ .

$lim_{t \to 0} \frac{t^{2} + t}{t (t^{2} + t)} - \frac{1}{t^{2} + t} \cdot \frac{t}{t}$

Step 1.1.3.2

Multiply $\frac{1}{t^{2} + t}$ by $\frac{t}{t}$ .

$lim_{t \to 0} \frac{t^{2} + t}{t (t^{2} + t)} - \frac{t}{(t^{2} + t) t}$

Step 1.1.3.3

Reorder the factors of $(t^{2} + t) t$ .

$lim_{t \to 0} \frac{t^{2} + t}{t (t^{2} + t)} - \frac{t}{t (t^{2} + t)}$

Step 1.1.4

Combine the numerators over the common denominator.

$lim_{t \to 0} \frac{t^{2} + t - t}{t (t^{2} + t)}$

Step 1.1.5

Subtract $t$ from $t$ .

$lim_{t \to 0} \frac{t^{2} + 0}{t (t^{2} + t)}$

Step 1.1.6

Add $t^{2}$ and $0$ .

$lim_{t \to 0} \frac{t^{2}}{t (t^{2} + t)}$

Step 1.2

Cancel the common factor of $t^{2}$ and $t$ .

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Step 1.2.1

Factor $t$ out of $t^{2}$ .

$lim_{t \to 0} \frac{t \cdot t}{t (t^{2} + t)}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Cancel the common factor.

$lim_{t \to 0} \frac{t \cdot t}{t (t^{2} + t)}$

Step 1.2.2.2

Rewrite the expression.

$lim_{t \to 0} \frac{t}{t^{2} + t}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{t \to 0} t}{lim_{t \to 0} t^{2} + t}$

Step 2.1.2

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{0}{lim_{t \to 0} t^{2} + t}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$\frac{0}{lim_{t \to 0} t^{2} + lim_{t \to 0} t}$

Step 2.1.3.2

Move the exponent $2$ from $t^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{t \to 0} t)}^{2} + lim_{t \to 0} t}$

Step 2.1.3.3

Evaluate the limits by plugging in $0$ for all occurrences of $t$ .

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Step 2.1.3.3.1

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{0}{0^{2} + lim_{t \to 0} t}$

Step 2.1.3.3.2

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{0}{0^{2} + 0}$

Step 2.1.3.4

Simplify the answer.

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Step 2.1.3.4.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0 + 0}$

Step 2.1.3.4.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 2.1.3.4.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.3.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{t \to 0} \frac{t}{t^{2} + t} = lim_{t \to 0} \frac{\frac{d}{d t} [t]}{\frac{d}{d t} [t^{2} + t]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{t \to 0} \frac{\frac{d}{d t} [t]}{\frac{d}{d t} [t^{2} + t]}$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 0} \frac{1}{\frac{d}{d t} [t^{2} + t]}$

Step 2.3.3

By the Sum Rule, the derivative of $t^{2} + t$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [t]$ .

$lim_{t \to 0} \frac{1}{\frac{d}{d t} [t^{2}] + \frac{d}{d t} [t]}$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$lim_{t \to 0} \frac{1}{2 t + \frac{d}{d t} [t]}$

Step 2.3.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$lim_{t \to 0} \frac{1}{2 t + 1}$

Step 3

Evaluate the limit.

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Step 3.1

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $0$ .

$\frac{lim_{t \to 0} 1}{lim_{t \to 0} 2 t + 1}$

Step 3.2

Evaluate the limit of $1$ which is constant as $t$ approaches $0$ .

$\frac{1}{lim_{t \to 0} 2 t + 1}$

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $0$ .

$\frac{1}{lim_{t \to 0} 2 t + lim_{t \to 0} 1}$

Step 3.4

Move the term $2$ outside of the limit because it is constant with respect to $t$ .

$\frac{1}{2 lim_{t \to 0} t + lim_{t \to 0} 1}$

Step 3.5

Evaluate the limit of $1$ which is constant as $t$ approaches $0$ .

$\frac{1}{2 lim_{t \to 0} t + 1}$

Step 4

Evaluate the limit of $t$ by plugging in $0$ for $t$ .

$\frac{1}{2 \cdot 0 + 1}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $2$ by $0$ .

$\frac{1}{0 + 1}$

Step 5.1.2

Add $0$ and $1$ .

$\frac{1}{1}$

Step 5.2

Divide $1$ by $1$ .

$1$