Calculus Examples

$h (x) = {\begin{cases} x^{2} + k^{2}, 0 \leq x \leq 2 \\ \frac{2 x + 4}{x - 1}, x > 2 \end{cases}$

Step 1

Find the limit of $\frac{2 x + 4}{x - 1}$ as $x$ approaches $2$ from the right.

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Step 1.1

Change the two-sided limit into a right sided limit.

$lim_{x \to 2^{+}} \frac{2 x + 4}{x - 1}$

Step 1.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2^{+}} 2 x + 4}{lim_{x \to 2^{+}} x - 1}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2^{+}} 2 x + lim_{x \to 2^{+}} 4}{lim_{x \to 2^{+}} x - 1}$

Step 1.4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 2^{+}} x + lim_{x \to 2^{+}} 4}{lim_{x \to 2^{+}} x - 1}$

Step 1.5

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{2 lim_{x \to 2^{+}} x + 4}{lim_{x \to 2^{+}} x - 1}$

Step 1.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{2 lim_{x \to 2^{+}} x + 4}{lim_{x \to 2^{+}} x - lim_{x \to 2^{+}} 1}$

Step 1.7

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

$\frac{2 lim_{x \to 2^{+}} x + 4}{lim_{x \to 2^{+}} x - 1 \cdot 1}$

Step 1.8

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 1.8.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2 \cdot 2 + 4}{lim_{x \to 2^{+}} x - 1 \cdot 1}$

Step 1.8.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2 \cdot 2 + 4}{2 - 1 \cdot 1}$

Step 1.9

Simplify the answer.

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Step 1.9.1

Simplify the numerator.

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Step 1.9.1.1

Multiply $2$ by $2$ .

$\frac{4 + 4}{2 - 1 \cdot 1}$

Step 1.9.1.2

Add $4$ and $4$ .

$\frac{8}{2 - 1 \cdot 1}$

Step 1.9.2

Simplify the denominator.

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Step 1.9.2.1

Multiply $- 1$ by $1$ .

$\frac{8}{2 - 1}$

Step 1.9.2.2

Subtract $1$ from $2$ .

$\frac{8}{1}$

Step 1.9.3

Divide $8$ by $1$ .

$8$

Step 2

Evaluate $x^{2} + k^{2}$ at $2$ .

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

${(2)}^{2} + k^{2}$

Step 2.2

Raise $2$ to the power of $2$ .

$4 + k^{2}$

Step 3

Since the limit of $\frac{2 x + 4}{x - 1}$ as $x$ approaches $2$ from the right is not equal to the function value at $x = 2$ , the function is not continuous at $x = 2$ .

Not continuous

Step 4