Calculus Examples

$lim_{h \to 0} \frac{1 - \frac{9}{h^{2}}}{1 - \frac{3}{h}}$

Step 1

Combine terms.

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Step 1.1

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

$lim_{h \to 0} \frac{\frac{h^{2}}{h^{2}} - \frac{9}{h^{2}}}{1 - \frac{3}{h}}$

Step 1.2

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{h^{2} - 9}{h^{2}}}{1 - \frac{3}{h}}$

Step 1.3

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

$lim_{h \to 0} \frac{\frac{h^{2} - 9}{h^{2}}}{\frac{h}{h} - \frac{3}{h}}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{\frac{h^{2} - 9}{h^{2}}}{\frac{h - 3}{h}}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{h^{2} - 9}{h^{2}} \cdot \frac{h}{h - 3}$

Step 2.2

Multiply $\frac{h^{2} - 9}{h^{2}}$ by $\frac{h}{h - 3}$ .

$lim_{h \to 0} \frac{(h^{2} - 9) h}{h^{2} (h - 3)}$

Step 2.3

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

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Step 2.3.1

Factor $h$ out of $(h^{2} - 9) h$ .

$lim_{h \to 0} \frac{h (h^{2} - 9)}{h^{2} (h - 3)}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $h$ out of $h^{2} (h - 3)$ .

$lim_{h \to 0} \frac{h (h^{2} - 9)}{h (h (h - 3))}$

Step 2.3.2.2

Cancel the common factor.

$lim_{h \to 0} \frac{h (h^{2} - 9)}{h (h (h - 3))}$

Step 2.3.2.3

Rewrite the expression.

$lim_{h \to 0} \frac{h^{2} - 9}{h (h - 3)}$

Step 3

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

$Does not exist$