Calculus Examples

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

Step 1

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

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

Step 2

Split the limit using the Product of Limits Rule on the limit as $h$ approaches $0$ .

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

Step 3

Move the exponent $2$ from ${(x + h)}^{2}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

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

Step 6

Move the limit inside the trig function because sine is continuous.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

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

Step 11

Evaluate the limit of $x^{2} \sin (\frac{1}{x})$ which is constant as $h$ approaches $0$ .

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

Step 12

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

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Step 12.1

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

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

Step 12.2

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

${(x + 0)}^{2} \cdot \sin (\frac{1}{x + 0}) - x^{2} \sin (\frac{1}{x})$

Step 13

Simplify the answer.

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Step 13.1

Combine the opposite terms in ${(x + 0)}^{2} \cdot \sin (\frac{1}{x + 0}) - x^{2} \sin (\frac{1}{x})$ .

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Step 13.1.1

Add $x$ and $0$ .

$x^{2} \cdot \sin (\frac{1}{x + 0}) - x^{2} \sin (\frac{1}{x})$

Step 13.1.2

Add $x$ and $0$ .

$x^{2} \cdot \sin (\frac{1}{x}) - x^{2} \sin (\frac{1}{x})$

Step 13.2

Subtract $x^{2} \sin (\frac{1}{x})$ from $x^{2} \sin (\frac{1}{x})$ .

$0$