Calculus Examples

$f (x) = {\begin{cases} x^{2} - c & x < 5 \\ 4 x + 2 c & x \geq 5 \end{cases}$

Step 1

Find the limit of $x^{2} - c$ as $x$ approaches $5$ from the left.

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Step 1.1

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

$lim_{x \to 5^{-}} x^{2} - c$

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 $5$ .

$lim_{x \to 5^{-}} x^{2} - lim_{x \to 5^{-}} c$

Step 1.2.2

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

${(lim_{x \to 5^{-}} x)}^{2} - lim_{x \to 5^{-}} c$

Step 1.2.3

Evaluate the limit of $c$ which is constant as $x$ approaches $5$ .

${(lim_{x \to 5^{-}} x)}^{2} - c$

Step 1.3

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$5^{2} - c$

Step 1.4

Raise $5$ to the power of $2$ .

$25 - c$

Step 2

Evaluate $4 x + 2 c$ at $5$ .

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Step 2.1

Replace the variable $x$ with $5$ in the expression.

$4 (5) + 2 c$

Step 2.2

Multiply $4$ by $5$ .

$20 + 2 c$

Step 3

Since the limit of $x^{2} - c$ as $x$ approaches $5$ from the left is not equal to the function value at $x = 5$ , the function is not continuous at $x = 5$ .

Not continuous

Step 4