Calculus Examples

$\frac{lim_{t \to 0} e^{3 t^{2} + 4 t}}{7 t} - lim_{t \to 0} \frac{1}{7 t}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit into the exponent.

$\frac{e^{lim_{t \to 0} 3 t^{2} + 4 t}}{7 t} - lim_{t \to 0} \frac{1}{7 t}$

Step 1.2

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

$\frac{e^{lim_{t \to 0} 3 t^{2} + lim_{t \to 0} 4 t}}{7 t} - lim_{t \to 0} \frac{1}{7 t}$

Step 1.3

Move the term $3$ outside of the limit because it is constant with respect to $t$ .

$\frac{e^{3 lim_{t \to 0} t^{2} + lim_{t \to 0} 4 t}}{7 t} - lim_{t \to 0} \frac{1}{7 t}$

Step 1.4

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

$\frac{e^{3 {(lim_{t \to 0} t)}^{2} + lim_{t \to 0} 4 t}}{7 t} - lim_{t \to 0} \frac{1}{7 t}$

Step 1.5

Move the term $4$ outside of the limit because it is constant with respect to $t$ .

$\frac{e^{3 {(lim_{t \to 0} t)}^{2} + 4 lim_{t \to 0} t}}{7 t} - lim_{t \to 0} \frac{1}{7 t}$

Step 2

Since the function approaches $- \infty$ from the left but $\infty$ from the right, the limit does not exist.

$\frac{e^{3 {(lim_{t \to 0} t)}^{2} + 4 lim_{t \to 0} t}}{7 t} - Does not exist$

Step 3

The limit does not exist.

$Does not exist$