Calculus Examples

$lim_{h \to 0} h (1 + \frac{1}{h^{2}})$

Step 1

Combine terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$lim_{h \to 0} h (\frac{h^{2}}{h^{2}} + \frac{1}{h^{2}})$

Step 1.2

Combine the numerators over the common denominator.

$lim_{h \to 0} h (\frac{h^{2} + 1}{h^{2}})$

Step 2

Set up the limit as a left-sided limit.

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

Step 3

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

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Step 3.1

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

$h (\frac{0^{2} + 1}{0^{2}})$

Step 3.2

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

$h (\frac{0^{2} + 1}{0})$

Step 3.3

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

$Does not exist$

Step 4

Set up the limit as a right-sided limit.

$lim_{h \to 0^{+}} h (\frac{h^{2} + 1}{h^{2}})$

Step 5

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

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Step 5.1

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

$h (\frac{0^{2} + 1}{0^{2}})$

Step 5.2

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

$h (\frac{0^{2} + 1}{0})$

Step 5.3

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

$Does not exist$

Step 6

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

$Does not exist$