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