Calculus Examples

$lim_{z \to 0} z (\frac{e^{z^{2}} - 1}{z^{2}})$

Step 1

Set up the limit as a left-sided limit.

$lim_{z \to 0^{-}} z (\frac{e^{z^{2}} - 1}{z^{2}})$

Step 2

Evaluate the limits by plugging in the value for the variable.

Tap for more steps...

Step 2.1

Evaluate the limit of $z (\frac{e^{z^{2}} - 1}{z^{2}})$ by plugging in $0$ for $z$ .

$z (\frac{e^{0^{2}} - 1}{0^{2}})$

Step 2.2

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

$z (\frac{e^{0^{2}} - 1}{0})$

Step 2.3

Since $z (\frac{e^{0^{2}} - 1}{0})$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

$lim_{z \to 0^{+}} z (\frac{e^{z^{2}} - 1}{z^{2}})$

Step 4

Evaluate the limits by plugging in the value for the variable.

Tap for more steps...

Step 4.1

Evaluate the limit of $z (\frac{e^{z^{2}} - 1}{z^{2}})$ by plugging in $0$ for $z$ .

$z (\frac{e^{0^{2}} - 1}{0^{2}})$

Step 4.2

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

$z (\frac{e^{0^{2}} - 1}{0})$

Step 4.3

Since $z (\frac{e^{0^{2}} - 1}{0})$ is undefined, the limit does not exist.

$Does not exist$

Step 5

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$