Calculus Examples

$f (x) = {\begin{cases} x^{2} + 3 & x < - 2 \\ 5 & x = - 2 \\ - 3 x + 1 & x > - 2 \end{cases}$

Step 1

Find the limit of

$x^{2} + 3$ as

$x$ approaches

$- 2$ from the left.

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Step 1.1

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

$lim_{x \to {(- 2)}^{-}} x^{2} + 3$

Step 1.2

Evaluate the limit.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$- 2$ .

$lim_{x \to {(- 2)}^{-}} x^{2} + lim_{x \to {(- 2)}^{-}} 3$

Step 1.2.2

Move the exponent

$2$ from

$x^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to {(- 2)}^{-}} x)}^{2} + lim_{x \to {(- 2)}^{-}} 3$

Step 1.2.3

Evaluate the limit of

$3$ which is constant as

$x$ approaches

$- 2$ .

${(lim_{x \to {(- 2)}^{-}} x)}^{2} + 3$

Step 1.3

Evaluate the limit of

$x$ by plugging in

$- 2$ for

$x$ .

${(- 2)}^{2} + 3$

Step 1.4

Simplify the answer.

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Step 1.4.1

Raise

$- 2$ to the power of

$2$ .

$4 + 3$

Step 1.4.2

Add

$4$ and

$3$ .

$7$

Step 2

Replace the variable

$x$ with

$- 2$ in the expression.

$5$

Step 3

Since the limit of

$x^{2} + 3$ as

$x$ approaches

$- 2$ from the left is not equal to the function value at

$x = - 2$ , the function is not continuous at

$x = - 2$ .

Not continuous

Step 4